| L(s) = 1 | + 1.40·5-s + 4.51·7-s − 6.07·11-s − 4.91·13-s − 0.433·17-s − 19-s − 0.995·23-s − 3.03·25-s + 2.23·29-s + 1.27·31-s + 6.31·35-s + 3.80·37-s + 5.31·41-s + 1.70·43-s − 9.71·47-s + 13.3·49-s − 4.70·53-s − 8.51·55-s − 8.18·59-s + 12.2·61-s − 6.88·65-s + 12.0·67-s + 9.35·71-s − 7.79·73-s − 27.4·77-s − 10.3·79-s − 10.5·83-s + ⋯ |
| L(s) = 1 | + 0.626·5-s + 1.70·7-s − 1.83·11-s − 1.36·13-s − 0.105·17-s − 0.229·19-s − 0.207·23-s − 0.607·25-s + 0.415·29-s + 0.229·31-s + 1.06·35-s + 0.625·37-s + 0.830·41-s + 0.260·43-s − 1.41·47-s + 1.90·49-s − 0.646·53-s − 1.14·55-s − 1.06·59-s + 1.57·61-s − 0.853·65-s + 1.47·67-s + 1.11·71-s − 0.911·73-s − 3.12·77-s − 1.16·79-s − 1.15·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 1.40T + 5T^{2} \) |
| 7 | \( 1 - 4.51T + 7T^{2} \) |
| 11 | \( 1 + 6.07T + 11T^{2} \) |
| 13 | \( 1 + 4.91T + 13T^{2} \) |
| 17 | \( 1 + 0.433T + 17T^{2} \) |
| 23 | \( 1 + 0.995T + 23T^{2} \) |
| 29 | \( 1 - 2.23T + 29T^{2} \) |
| 31 | \( 1 - 1.27T + 31T^{2} \) |
| 37 | \( 1 - 3.80T + 37T^{2} \) |
| 41 | \( 1 - 5.31T + 41T^{2} \) |
| 43 | \( 1 - 1.70T + 43T^{2} \) |
| 47 | \( 1 + 9.71T + 47T^{2} \) |
| 53 | \( 1 + 4.70T + 53T^{2} \) |
| 59 | \( 1 + 8.18T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 - 12.0T + 67T^{2} \) |
| 71 | \( 1 - 9.35T + 71T^{2} \) |
| 73 | \( 1 + 7.79T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + 5.84T + 89T^{2} \) |
| 97 | \( 1 - 1.47T + 97T^{2} \) |
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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
−7.74202956574646015300436542346, −6.90385829465246763432104527941, −5.87719214060592953428584699340, −5.21110859968465922670463925484, −4.89327472570764189140461286276, −4.13336104208746430612426405191, −2.64768513023979003205499484970, −2.36507443033027796195479622851, −1.43047361805966256365616795650, 0,
1.43047361805966256365616795650, 2.36507443033027796195479622851, 2.64768513023979003205499484970, 4.13336104208746430612426405191, 4.89327472570764189140461286276, 5.21110859968465922670463925484, 5.87719214060592953428584699340, 6.90385829465246763432104527941, 7.74202956574646015300436542346
