| L(s) = 1 | + 3·3-s − 2·5-s − 32·7-s + 9·9-s − 68·11-s + 13·13-s − 6·15-s − 14·17-s + 4·19-s − 96·21-s + 72·23-s − 121·25-s + 27·27-s + 102·29-s − 136·31-s − 204·33-s + 64·35-s − 386·37-s + 39·39-s + 250·41-s − 140·43-s − 18·45-s − 296·47-s + 681·49-s − 42·51-s + 526·53-s + 136·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.178·5-s − 1.72·7-s + 1/3·9-s − 1.86·11-s + 0.277·13-s − 0.103·15-s − 0.199·17-s + 0.0482·19-s − 0.997·21-s + 0.652·23-s − 0.967·25-s + 0.192·27-s + 0.653·29-s − 0.787·31-s − 1.07·33-s + 0.309·35-s − 1.71·37-s + 0.160·39-s + 0.952·41-s − 0.496·43-s − 0.0596·45-s − 0.918·47-s + 1.98·49-s − 0.115·51-s + 1.36·53-s + 0.333·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 13 | \( 1 - p T \) |
| good | 5 | \( 1 + 2 T + p^{3} T^{2} \) |
| 7 | \( 1 + 32 T + p^{3} T^{2} \) |
| 11 | \( 1 + 68 T + p^{3} T^{2} \) |
| 17 | \( 1 + 14 T + p^{3} T^{2} \) |
| 19 | \( 1 - 4 T + p^{3} T^{2} \) |
| 23 | \( 1 - 72 T + p^{3} T^{2} \) |
| 29 | \( 1 - 102 T + p^{3} T^{2} \) |
| 31 | \( 1 + 136 T + p^{3} T^{2} \) |
| 37 | \( 1 + 386 T + p^{3} T^{2} \) |
| 41 | \( 1 - 250 T + p^{3} T^{2} \) |
| 43 | \( 1 + 140 T + p^{3} T^{2} \) |
| 47 | \( 1 + 296 T + p^{3} T^{2} \) |
| 53 | \( 1 - 526 T + p^{3} T^{2} \) |
| 59 | \( 1 - 332 T + p^{3} T^{2} \) |
| 61 | \( 1 + 410 T + p^{3} T^{2} \) |
| 67 | \( 1 - 596 T + p^{3} T^{2} \) |
| 71 | \( 1 + 880 T + p^{3} T^{2} \) |
| 73 | \( 1 - 506 T + p^{3} T^{2} \) |
| 79 | \( 1 + 640 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1380 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1450 T + p^{3} T^{2} \) |
| 97 | \( 1 + 446 T + p^{3} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.26269677708148445043240901361, −10.68705199540404301049261338994, −9.945577606556881118331020590802, −8.916015699292289699565810734580, −7.76413033339616063583774718820, −6.70415960999532286066369518848, −5.37082396999903280422523477009, −3.62580570997257086065486485682, −2.60627818107201396350141935411, 0,
2.60627818107201396350141935411, 3.62580570997257086065486485682, 5.37082396999903280422523477009, 6.70415960999532286066369518848, 7.76413033339616063583774718820, 8.916015699292289699565810734580, 9.945577606556881118331020590802, 10.68705199540404301049261338994, 12.26269677708148445043240901361
