| L(s) = 1 | − 14·3-s + 34·5-s − 98·7-s − 278·9-s + 420·11-s − 490·13-s − 476·15-s − 1.05e3·17-s + 1.24e3·19-s + 1.37e3·21-s − 504·23-s − 2.39e3·25-s + 7.12e3·27-s − 3.90e3·29-s + 2.04e3·31-s − 5.88e3·33-s − 3.33e3·35-s − 7.48e3·37-s + 6.86e3·39-s + 7.83e3·41-s + 1.03e4·43-s − 9.45e3·45-s + 4.19e4·47-s + 7.20e3·49-s + 1.47e4·51-s + 3.28e4·53-s + 1.42e4·55-s + ⋯ |
| L(s) = 1 | − 0.898·3-s + 0.608·5-s − 0.755·7-s − 1.14·9-s + 1.04·11-s − 0.804·13-s − 0.546·15-s − 0.886·17-s + 0.791·19-s + 0.678·21-s − 0.198·23-s − 0.766·25-s + 1.88·27-s − 0.862·29-s + 0.382·31-s − 0.939·33-s − 0.459·35-s − 0.899·37-s + 0.722·39-s + 0.727·41-s + 0.852·43-s − 0.695·45-s + 2.77·47-s + 3/7·49-s + 0.795·51-s + 1.60·53-s + 0.636·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50176 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50176 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.089226190\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.089226190\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + 14 T + 158 p T^{2} + 14 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 34 T + 142 p^{2} T^{2} - 34 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 420 T + 237126 T^{2} - 420 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 490 T + 656150 T^{2} + 490 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1056 T + 3106542 T^{2} + 1056 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 1246 T + 1618778 T^{2} - 1246 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 504 T + 12920574 T^{2} + 504 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 3904 T + 26647526 T^{2} + 3904 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2044 T + 33160782 T^{2} - 2044 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 7488 T + 152693494 T^{2} + 7488 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 7832 T + 168604142 T^{2} - 7832 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 10332 T + 248230342 T^{2} - 10332 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 41972 T + 855029710 T^{2} - 41972 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 32812 T + 664801838 T^{2} - 32812 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 48398 T + 1851574618 T^{2} - 48398 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 718 T + 1677458142 T^{2} + 718 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 12824 T + 1908078582 T^{2} - 12824 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 103992 T + 6302476942 T^{2} - 103992 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 54100 T + 3096449510 T^{2} + 54100 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 64568 T + 7121481950 T^{2} - 64568 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 47810 T + 8432588842 T^{2} + 47810 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 17388 T - 1489722410 T^{2} + 17388 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 97296 T + 17237136142 T^{2} + 97296 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−11.45319670923299355643925732946, −11.41276710441196694916770207739, −10.63244333028578195602037247401, −10.27894901779748889729001180206, −9.569369161247496113401394444188, −9.304338076822741088434643129833, −8.811361051441980219871438239035, −8.318899304455427856038960175580, −7.25635983127309522821361071813, −7.17070242705292405579710771628, −6.21307047659836030599724815678, −6.11897209167785143791704398550, −5.41360389524704514500132330249, −5.15925342411593564007758354475, −4.03358900424641313592678229534, −3.70483769102511447093288448136, −2.45598976059752256448345259672, −2.40804573130240669842264132597, −1.02793582829200226071371025714, −0.38130564166366943337714664698,
0.38130564166366943337714664698, 1.02793582829200226071371025714, 2.40804573130240669842264132597, 2.45598976059752256448345259672, 3.70483769102511447093288448136, 4.03358900424641313592678229534, 5.15925342411593564007758354475, 5.41360389524704514500132330249, 6.11897209167785143791704398550, 6.21307047659836030599724815678, 7.17070242705292405579710771628, 7.25635983127309522821361071813, 8.318899304455427856038960175580, 8.811361051441980219871438239035, 9.304338076822741088434643129833, 9.569369161247496113401394444188, 10.27894901779748889729001180206, 10.63244333028578195602037247401, 11.41276710441196694916770207739, 11.45319670923299355643925732946
