| L(s) = 1 | + 2·2-s + 3·4-s + 2·7-s + 4·8-s − 4·13-s + 4·14-s + 5·16-s + 8·17-s + 8·19-s − 2·23-s − 8·26-s + 6·28-s + 4·29-s − 4·31-s + 6·32-s + 16·34-s − 4·37-s + 16·38-s − 6·41-s + 16·43-s − 4·46-s + 16·47-s + 3·49-s − 12·52-s + 8·56-s + 8·58-s + 4·59-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 0.755·7-s + 1.41·8-s − 1.10·13-s + 1.06·14-s + 5/4·16-s + 1.94·17-s + 1.83·19-s − 0.417·23-s − 1.56·26-s + 1.13·28-s + 0.742·29-s − 0.718·31-s + 1.06·32-s + 2.74·34-s − 0.657·37-s + 2.59·38-s − 0.937·41-s + 2.43·43-s − 0.589·46-s + 2.33·47-s + 3/7·49-s − 1.66·52-s + 1.06·56-s + 1.05·58-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89302500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89302500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(14.01168928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.01168928\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 11 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 51 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 35 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 50 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 63 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 16 T + 138 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 16 T + 146 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 175 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 20 T + 198 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 114 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 190 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 146 T^{2} + p^{2} T^{4} \) |
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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
−7.48547119668401162695854659721, −7.42060253230240032618750730426, −7.28206586646266552736610757147, −7.10906745054166816016678961654, −6.15666095172930013156633412714, −6.11424666223687944322368273447, −5.60407237489492428633729024777, −5.54626575236342589924684132457, −5.02536360860568827844839971637, −4.91194540177050100715608673949, −4.42252992477271427311391874327, −4.05066867099969081350829564487, −3.65557105700816608550015976629, −3.30074612997481487973740371713, −2.80928999505812828281901754044, −2.71863138031296280019903577088, −1.90922888309098944470021687333, −1.79647686283815860656325940497, −0.916043868501342753699962991786, −0.794369016615572623477208408104,
0.794369016615572623477208408104, 0.916043868501342753699962991786, 1.79647686283815860656325940497, 1.90922888309098944470021687333, 2.71863138031296280019903577088, 2.80928999505812828281901754044, 3.30074612997481487973740371713, 3.65557105700816608550015976629, 4.05066867099969081350829564487, 4.42252992477271427311391874327, 4.91194540177050100715608673949, 5.02536360860568827844839971637, 5.54626575236342589924684132457, 5.60407237489492428633729024777, 6.11424666223687944322368273447, 6.15666095172930013156633412714, 7.10906745054166816016678961654, 7.28206586646266552736610757147, 7.42060253230240032618750730426, 7.48547119668401162695854659721
