L(s) = 1 | + 3-s − 2·4-s + 7-s + 9-s − 11-s − 2·12-s + 5·13-s + 4·16-s + 3·17-s + 5·19-s + 21-s + 27-s − 2·28-s + 6·29-s − 4·31-s − 33-s − 2·36-s − 7·37-s + 5·39-s + 3·41-s − 4·43-s + 2·44-s − 12·47-s + 4·48-s + 49-s + 3·51-s − 10·52-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 4-s + 0.377·7-s + 1/3·9-s − 0.301·11-s − 0.577·12-s + 1.38·13-s + 16-s + 0.727·17-s + 1.14·19-s + 0.218·21-s + 0.192·27-s − 0.377·28-s + 1.11·29-s − 0.718·31-s − 0.174·33-s − 1/3·36-s − 1.15·37-s + 0.800·39-s + 0.468·41-s − 0.609·43-s + 0.301·44-s − 1.75·47-s + 0.577·48-s + 1/7·49-s + 0.420·51-s − 1.38·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.311683428\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.311683428\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−8.309929758221887858916490616743, −7.62753502779983703960438753675, −6.80205578078937662469645174155, −5.75321570936245125213987808376, −5.22933907909754573008137333310, −4.41661865701975028475130207902, −3.55538076162676652379857487945, −3.11780086777596111077544565896, −1.69555021853745340309263897637, −0.851153672245461197326133665671,
0.851153672245461197326133665671, 1.69555021853745340309263897637, 3.11780086777596111077544565896, 3.55538076162676652379857487945, 4.41661865701975028475130207902, 5.22933907909754573008137333310, 5.75321570936245125213987808376, 6.80205578078937662469645174155, 7.62753502779983703960438753675, 8.309929758221887858916490616743