L(s) = 1 | + 3-s − 3·5-s − 3·7-s + 9-s + 3·11-s − 3·15-s + 17-s + 19-s − 3·21-s + 4·25-s + 27-s + 8·29-s + 2·31-s + 3·33-s + 9·35-s + 4·37-s − 12·41-s − 43-s − 3·45-s − 9·47-s + 2·49-s + 51-s − 6·53-s − 9·55-s + 57-s − 6·59-s + 61-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 1.34·5-s − 1.13·7-s + 1/3·9-s + 0.904·11-s − 0.774·15-s + 0.242·17-s + 0.229·19-s − 0.654·21-s + 4/5·25-s + 0.192·27-s + 1.48·29-s + 0.359·31-s + 0.522·33-s + 1.52·35-s + 0.657·37-s − 1.87·41-s − 0.152·43-s − 0.447·45-s − 1.31·47-s + 2/7·49-s + 0.140·51-s − 0.824·53-s − 1.21·55-s + 0.132·57-s − 0.781·59-s + 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.253156499371799472511386458956, −7.47383809956106347393899504338, −6.73345911707136490794401067909, −6.23704271046523504270082085925, −4.88148310109735790089290733133, −4.13570499293752999790440740743, −3.37888131212954161008514599002, −2.91391913505672031423624549582, −1.34792821524806844044689768710, 0,
1.34792821524806844044689768710, 2.91391913505672031423624549582, 3.37888131212954161008514599002, 4.13570499293752999790440740743, 4.88148310109735790089290733133, 6.23704271046523504270082085925, 6.73345911707136490794401067909, 7.47383809956106347393899504338, 8.253156499371799472511386458956