| L(s) = 1 | + 7·2-s − 3·3-s + 25·4-s + 10·5-s − 21·6-s + 25·7-s + 63·8-s − 43·9-s + 70·10-s − 75·12-s + 50·13-s + 175·14-s − 30·15-s + 169·16-s + 151·17-s − 301·18-s + 3·19-s + 250·20-s − 75·21-s + 48·23-s − 189·24-s + 75·25-s + 350·26-s + 204·27-s + 625·28-s + 221·29-s − 210·30-s + ⋯ |
| L(s) = 1 | + 2.47·2-s − 0.577·3-s + 25/8·4-s + 0.894·5-s − 1.42·6-s + 1.34·7-s + 2.78·8-s − 1.59·9-s + 2.21·10-s − 1.80·12-s + 1.06·13-s + 3.34·14-s − 0.516·15-s + 2.64·16-s + 2.15·17-s − 3.94·18-s + 0.0362·19-s + 2.79·20-s − 0.779·21-s + 0.435·23-s − 1.60·24-s + 3/5·25-s + 2.64·26-s + 1.45·27-s + 4.21·28-s + 1.41·29-s − 1.27·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 366025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(17.74878308\) |
| \(L(\frac12)\) |
\(\approx\) |
\(17.74878308\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - 7 T + 3 p^{3} T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + p T + 52 T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 25 T + 498 T^{2} - 25 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 50 T + 4594 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 151 T + 14298 T^{2} - 151 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 3 T + 5114 T^{2} - 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 48 T + 24842 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 221 T + 39564 T^{2} - 221 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 141 T + 6374 T^{2} - 141 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 559 T + 171568 T^{2} + 559 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 144 T + 39598 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 60 T + 56486 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 48 T + 3526 p T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 117 T + 179792 T^{2} - 117 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 86 T + 306510 T^{2} + 86 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 155 T + 399784 T^{2} - 155 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 266 T + 523590 T^{2} - 266 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1587 T + 1329650 T^{2} - 1587 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 70 T + 764962 T^{2} + 70 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1294 T + 1349454 T^{2} - 1294 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 558 T + 911590 T^{2} - 558 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 1777 T + 2191512 T^{2} + 1777 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 334 T + 977242 T^{2} + 334 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.60970007863868388447273568818, −10.40335165362817584604611235140, −9.845572987424079976685145169298, −9.113300283836949239862552560290, −8.502349391220599004124457374645, −8.128990925953505199932798244850, −7.932210956650347655102640537734, −6.99826159061041958469502136704, −6.30553913608353578590922633673, −6.28418218917552830209031942346, −5.49588942808232872576462206606, −5.39650823513097301451742651166, −5.05517564821020346452831397066, −4.74930780343235686892296600621, −3.73946661765320663390552653761, −3.54722751194479567668834270929, −2.86539025584696441024776349500, −2.38676096773606048045772383350, −1.33582996965328396768142984703, −0.961052795209085454789672262332,
0.961052795209085454789672262332, 1.33582996965328396768142984703, 2.38676096773606048045772383350, 2.86539025584696441024776349500, 3.54722751194479567668834270929, 3.73946661765320663390552653761, 4.74930780343235686892296600621, 5.05517564821020346452831397066, 5.39650823513097301451742651166, 5.49588942808232872576462206606, 6.28418218917552830209031942346, 6.30553913608353578590922633673, 6.99826159061041958469502136704, 7.932210956650347655102640537734, 8.128990925953505199932798244850, 8.502349391220599004124457374645, 9.113300283836949239862552560290, 9.845572987424079976685145169298, 10.40335165362817584604611235140, 10.60970007863868388447273568818
