| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s − 6·6-s − 8·7-s − 8·8-s + 9·9-s + 12·11-s + 12·12-s − 13·13-s + 16·14-s + 16·16-s + 42·17-s − 18·18-s − 52·19-s − 24·21-s − 24·22-s − 132·23-s − 24·24-s + 26·26-s + 27·27-s − 32·28-s + 282·29-s + 116·31-s − 32·32-s + 36·33-s − 84·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.431·7-s − 0.353·8-s + 1/3·9-s + 0.328·11-s + 0.288·12-s − 0.277·13-s + 0.305·14-s + 1/4·16-s + 0.599·17-s − 0.235·18-s − 0.627·19-s − 0.249·21-s − 0.232·22-s − 1.19·23-s − 0.204·24-s + 0.196·26-s + 0.192·27-s − 0.215·28-s + 1.80·29-s + 0.672·31-s − 0.176·32-s + 0.189·33-s − 0.423·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + p T \) |
| good | 7 | \( 1 + 8 T + p^{3} T^{2} \) |
| 11 | \( 1 - 12 T + p^{3} T^{2} \) |
| 17 | \( 1 - 42 T + p^{3} T^{2} \) |
| 19 | \( 1 + 52 T + p^{3} T^{2} \) |
| 23 | \( 1 + 132 T + p^{3} T^{2} \) |
| 29 | \( 1 - 282 T + p^{3} T^{2} \) |
| 31 | \( 1 - 116 T + p^{3} T^{2} \) |
| 37 | \( 1 + 398 T + p^{3} T^{2} \) |
| 41 | \( 1 - 174 T + p^{3} T^{2} \) |
| 43 | \( 1 - 76 T + p^{3} T^{2} \) |
| 47 | \( 1 + 456 T + p^{3} T^{2} \) |
| 53 | \( 1 + 150 T + p^{3} T^{2} \) |
| 59 | \( 1 + 156 T + p^{3} T^{2} \) |
| 61 | \( 1 - 230 T + p^{3} T^{2} \) |
| 67 | \( 1 - 592 T + p^{3} T^{2} \) |
| 71 | \( 1 - 408 T + p^{3} T^{2} \) |
| 73 | \( 1 - 10 p T + p^{3} T^{2} \) |
| 79 | \( 1 - 728 T + p^{3} T^{2} \) |
| 83 | \( 1 + 36 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1482 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1742 T + p^{3} T^{2} \) |
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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
−8.211948495259929922878626513705, −8.096821651337592044261560039279, −6.80017974243895666242993112137, −6.45393667091049577166082726140, −5.24969369783779559601322372738, −4.14018972680080869011980637118, −3.21500100262916492739981769990, −2.31884629445748682865244562239, −1.26645166178808878006035466198, 0,
1.26645166178808878006035466198, 2.31884629445748682865244562239, 3.21500100262916492739981769990, 4.14018972680080869011980637118, 5.24969369783779559601322372738, 6.45393667091049577166082726140, 6.80017974243895666242993112137, 8.096821651337592044261560039279, 8.211948495259929922878626513705
