| L(s) = 1 | − 19·2-s + 45·3-s + 99·4-s − 353·5-s − 855·6-s − 2.00e3·7-s + 665·8-s − 2.09e3·9-s + 6.70e3·10-s − 1.81e3·11-s + 4.45e3·12-s + 4.39e3·13-s + 3.81e4·14-s − 1.58e4·15-s − 8.74e3·16-s − 2.53e4·17-s + 3.98e4·18-s + 2.21e4·19-s − 3.49e4·20-s − 9.04e4·21-s + 3.43e4·22-s − 2.64e4·23-s + 2.99e4·24-s − 5.25e4·25-s − 8.34e4·26-s − 1.81e5·27-s − 1.98e5·28-s + ⋯ |
| L(s) = 1 | − 1.67·2-s + 0.962·3-s + 0.773·4-s − 1.26·5-s − 1.61·6-s − 2.21·7-s + 0.459·8-s − 0.958·9-s + 2.12·10-s − 0.410·11-s + 0.744·12-s + 0.554·13-s + 3.71·14-s − 1.21·15-s − 0.533·16-s − 1.25·17-s + 1.61·18-s + 0.739·19-s − 0.976·20-s − 2.13·21-s + 0.688·22-s − 0.452·23-s + 0.441·24-s − 0.673·25-s − 0.931·26-s − 1.77·27-s − 1.71·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + 19 T + 131 p T^{2} + 19 p^{7} T^{3} + p^{14} T^{4} \) |
| 3 | $D_{4}$ | \( 1 - 5 p^{2} T + 458 p^{2} T^{2} - 5 p^{9} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 353 T + 177208 T^{2} + 353 p^{7} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 41 p^{2} T + 2649282 T^{2} + 41 p^{9} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 1810 T + 7198390 T^{2} + 1810 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 25361 T + 643460668 T^{2} + 25361 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 22106 T + 141106790 T^{2} - 22106 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 26424 T + 3097648846 T^{2} + 26424 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5804 T + 33449999614 T^{2} + 5804 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 39744 T + 54390419758 T^{2} - 39744 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 163299 T + 181994892160 T^{2} - 163299 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8070 p T + 274020654610 T^{2} + 8070 p^{8} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 229307 T + 555183291698 T^{2} - 229307 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1638525 T + 1682936215978 T^{2} + 1638525 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 1046382 T + 1701182112730 T^{2} - 1046382 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 370158 T + 4837245961366 T^{2} + 370158 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4675422 T + 11232926947738 T^{2} - 4675422 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1821402 T + 6715905784390 T^{2} + 1821402 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1135611 T - 6597281929502 T^{2} + 1135611 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 6459284 T + 32251656236358 T^{2} + 6459284 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 73808 T + 15935358386334 T^{2} + 73808 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12100972 T + 88634017455958 T^{2} + 12100972 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 9815060 T + 78068771423926 T^{2} - 9815060 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 17591688 T + 206995518168430 T^{2} + 17591688 p^{7} T^{3} + p^{14} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.96231838979939430214362877960, −17.53230841807185624016288026070, −16.44289600219347302641427970071, −16.05785300841779083825692080933, −15.54274675964509805113350700552, −14.63751532505320040554820064276, −13.36050850790501114076326534209, −13.31500267367336370374631730363, −11.91143787794144075801944846525, −11.20028759746125557676131326373, −9.856593243082484925163368262153, −9.618604078505776560455158453639, −8.522882492990170670482305213004, −8.414964274658800881773517699215, −7.32757064790059812258803009475, −6.11135350154495572449639728437, −3.78827778364534837651313235021, −2.89805707990915307248957329406, 0, 0,
2.89805707990915307248957329406, 3.78827778364534837651313235021, 6.11135350154495572449639728437, 7.32757064790059812258803009475, 8.414964274658800881773517699215, 8.522882492990170670482305213004, 9.618604078505776560455158453639, 9.856593243082484925163368262153, 11.20028759746125557676131326373, 11.91143787794144075801944846525, 13.31500267367336370374631730363, 13.36050850790501114076326534209, 14.63751532505320040554820064276, 15.54274675964509805113350700552, 16.05785300841779083825692080933, 16.44289600219347302641427970071, 17.53230841807185624016288026070, 17.96231838979939430214362877960
