| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s − 5·5-s − 6·6-s − 16·7-s − 8·8-s + 9·9-s + 10·10-s + 42·11-s + 12·12-s + 44·13-s + 32·14-s − 15·15-s + 16·16-s − 42·17-s − 18·18-s − 52·19-s − 20·20-s − 48·21-s − 84·22-s + 23·23-s − 24·24-s + 25·25-s − 88·26-s + 27·27-s − 64·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.863·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.15·11-s + 0.288·12-s + 0.938·13-s + 0.610·14-s − 0.258·15-s + 1/4·16-s − 0.599·17-s − 0.235·18-s − 0.627·19-s − 0.223·20-s − 0.498·21-s − 0.814·22-s + 0.208·23-s − 0.204·24-s + 1/5·25-s − 0.663·26-s + 0.192·27-s − 0.431·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.500299530\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.500299530\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 + p T \) |
| 23 | \( 1 - p T \) |
| good | 7 | \( 1 + 16 T + p^{3} T^{2} \) |
| 11 | \( 1 - 42 T + p^{3} T^{2} \) |
| 13 | \( 1 - 44 T + p^{3} T^{2} \) |
| 17 | \( 1 + 42 T + p^{3} T^{2} \) |
| 19 | \( 1 + 52 T + p^{3} T^{2} \) |
| 29 | \( 1 + 108 T + p^{3} T^{2} \) |
| 31 | \( 1 + 160 T + p^{3} T^{2} \) |
| 37 | \( 1 - 146 T + p^{3} T^{2} \) |
| 41 | \( 1 - 6 T + p^{3} T^{2} \) |
| 43 | \( 1 - 302 T + p^{3} T^{2} \) |
| 47 | \( 1 - 276 T + p^{3} T^{2} \) |
| 53 | \( 1 - 678 T + p^{3} T^{2} \) |
| 59 | \( 1 - 312 T + p^{3} T^{2} \) |
| 61 | \( 1 - 182 T + p^{3} T^{2} \) |
| 67 | \( 1 + 226 T + p^{3} T^{2} \) |
| 71 | \( 1 - 726 T + p^{3} T^{2} \) |
| 73 | \( 1 - 206 T + p^{3} T^{2} \) |
| 79 | \( 1 + 568 T + p^{3} T^{2} \) |
| 83 | \( 1 - 492 T + p^{3} T^{2} \) |
| 89 | \( 1 + 264 T + p^{3} T^{2} \) |
| 97 | \( 1 + 412 T + p^{3} T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.866329831570272522064242450081, −8.999349578897345608950665241753, −8.676097622311026274282342892075, −7.48625860142900528551034356622, −6.73397208630075670780765238805, −5.92081615470556135363820997471, −4.14213385695162509249914803703, −3.46521693864456222751603728527, −2.14343057278567401281126949387, −0.76758545918901127192967939579,
0.76758545918901127192967939579, 2.14343057278567401281126949387, 3.46521693864456222751603728527, 4.14213385695162509249914803703, 5.92081615470556135363820997471, 6.73397208630075670780765238805, 7.48625860142900528551034356622, 8.676097622311026274282342892075, 8.999349578897345608950665241753, 9.866329831570272522064242450081
