| L(s) = 1 | − 4·2-s − 3·3-s + 12·4-s + 10·5-s + 12·6-s − 17·7-s − 32·8-s − 29·9-s − 40·10-s + 18·11-s − 36·12-s − 9·13-s + 68·14-s − 30·15-s + 80·16-s − 79·17-s + 116·18-s + 34·19-s + 120·20-s + 51·21-s − 72·22-s − 46·23-s + 96·24-s + 75·25-s + 36·26-s + 120·27-s − 204·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 0.577·3-s + 3/2·4-s + 0.894·5-s + 0.816·6-s − 0.917·7-s − 1.41·8-s − 1.07·9-s − 1.26·10-s + 0.493·11-s − 0.866·12-s − 0.192·13-s + 1.29·14-s − 0.516·15-s + 5/4·16-s − 1.12·17-s + 1.51·18-s + 0.410·19-s + 1.34·20-s + 0.529·21-s − 0.697·22-s − 0.417·23-s + 0.816·24-s + 3/5·25-s + 0.271·26-s + 0.855·27-s − 1.37·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52900 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 23 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + p T + 38 T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 17 T + 740 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 18 T + 918 T^{2} - 18 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 9 T + 2936 T^{2} + 9 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 79 T + 9178 T^{2} + 79 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 34 T + 12182 T^{2} - 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 112 T + 51841 T^{2} + 112 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 92 T + 49361 T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 491 T + 160682 T^{2} + 491 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 332 T + 65461 T^{2} + 332 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 354 T + 181510 T^{2} + 354 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 599 T + 296890 T^{2} + 599 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 305 T + 320116 T^{2} + 305 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 357 T + 241414 T^{2} - 357 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 172 T + 458730 T^{2} + 172 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 531 T + 641338 T^{2} + 531 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1254 T + 911559 T^{2} - 1254 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 343 T + 797792 T^{2} + 343 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 88 T + 930782 T^{2} + 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1273 T + 1298882 T^{2} - 1273 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1106 T + 1709834 T^{2} - 1106 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2240 T + 2588894 T^{2} + 2240 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−11.32277614476745674035118807367, −11.06261969262693208466955536733, −10.33377867759958170136745116282, −10.10890232067053498085827224232, −9.371211236571410539328900317643, −9.341279733082870591753859791922, −8.581279755219757847558971592284, −8.350720908631529229360829593159, −7.53128142988758434896139068843, −6.77430176058853290361448310050, −6.44591785665238699526585567015, −6.26064044448570022947235220788, −5.26476880772818464526443845018, −5.08285235786245020151215987983, −3.54923107364091131856081569418, −3.14432059586994515024987533966, −2.11678931097840996637049498027, −1.56218787078874294915871330238, 0, 0,
1.56218787078874294915871330238, 2.11678931097840996637049498027, 3.14432059586994515024987533966, 3.54923107364091131856081569418, 5.08285235786245020151215987983, 5.26476880772818464526443845018, 6.26064044448570022947235220788, 6.44591785665238699526585567015, 6.77430176058853290361448310050, 7.53128142988758434896139068843, 8.350720908631529229360829593159, 8.581279755219757847558971592284, 9.341279733082870591753859791922, 9.371211236571410539328900317643, 10.10890232067053498085827224232, 10.33377867759958170136745116282, 11.06261969262693208466955536733, 11.32277614476745674035118807367
