| L(s) = 1 | + 5-s − 11-s + 6·13-s + 3·17-s − 5·19-s − 2·23-s + 25-s + 5·29-s + 5·31-s − 37-s − 2·41-s − 12·43-s + 2·47-s + 13·53-s − 55-s − 2·59-s − 61-s + 6·65-s − 16·67-s + 15·71-s − 10·73-s − 2·79-s + 14·83-s + 3·85-s + 9·89-s − 5·95-s + 16·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.301·11-s + 1.66·13-s + 0.727·17-s − 1.14·19-s − 0.417·23-s + 1/5·25-s + 0.928·29-s + 0.898·31-s − 0.164·37-s − 0.312·41-s − 1.82·43-s + 0.291·47-s + 1.78·53-s − 0.134·55-s − 0.260·59-s − 0.128·61-s + 0.744·65-s − 1.95·67-s + 1.78·71-s − 1.17·73-s − 0.225·79-s + 1.53·83-s + 0.325·85-s + 0.953·89-s − 0.512·95-s + 1.62·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.265622788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.265622788\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 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
−12.37376689777929, −12.04958598466825, −11.58724975345135, −11.07074801567081, −10.49879211744283, −10.26570359015012, −9.976130838582578, −9.173940944349510, −8.764391125722456, −8.406962527089457, −8.053910685455655, −7.465647113558932, −6.782645323685110, −6.375288943655654, −6.081596904166403, −5.562474643818775, −4.972516892558027, −4.510555845678342, −3.871252191897000, −3.457785352527740, −2.890146115633761, −2.268500049948171, −1.676746059025951, −1.138706479190363, −0.4887467492212153,
0.4887467492212153, 1.138706479190363, 1.676746059025951, 2.268500049948171, 2.890146115633761, 3.457785352527740, 3.871252191897000, 4.510555845678342, 4.972516892558027, 5.562474643818775, 6.081596904166403, 6.375288943655654, 6.782645323685110, 7.465647113558932, 8.053910685455655, 8.406962527089457, 8.764391125722456, 9.173940944349510, 9.976130838582578, 10.26570359015012, 10.49879211744283, 11.07074801567081, 11.58724975345135, 12.04958598466825, 12.37376689777929
