L(s) = 1 | − 0.744·2-s − 3-s − 1.44·4-s + 1.71·5-s + 0.744·6-s − 3.69·7-s + 2.56·8-s + 9-s − 1.27·10-s − 0.320·11-s + 1.44·12-s − 1.38·13-s + 2.74·14-s − 1.71·15-s + 0.983·16-s + 17-s − 0.744·18-s − 1.35·19-s − 2.47·20-s + 3.69·21-s + 0.238·22-s + 3.98·23-s − 2.56·24-s − 2.05·25-s + 1.02·26-s − 27-s + 5.33·28-s + ⋯ |
L(s) = 1 | − 0.526·2-s − 0.577·3-s − 0.723·4-s + 0.766·5-s + 0.303·6-s − 1.39·7-s + 0.906·8-s + 0.333·9-s − 0.403·10-s − 0.0967·11-s + 0.417·12-s − 0.383·13-s + 0.734·14-s − 0.442·15-s + 0.245·16-s + 0.242·17-s − 0.175·18-s − 0.311·19-s − 0.554·20-s + 0.805·21-s + 0.0508·22-s + 0.831·23-s − 0.523·24-s − 0.411·25-s + 0.201·26-s − 0.192·27-s + 1.00·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 - T \) |
good | 2 | \( 1 + 0.744T + 2T^{2} \) |
| 5 | \( 1 - 1.71T + 5T^{2} \) |
| 7 | \( 1 + 3.69T + 7T^{2} \) |
| 11 | \( 1 + 0.320T + 11T^{2} \) |
| 13 | \( 1 + 1.38T + 13T^{2} \) |
| 19 | \( 1 + 1.35T + 19T^{2} \) |
| 23 | \( 1 - 3.98T + 23T^{2} \) |
| 29 | \( 1 - 3.50T + 29T^{2} \) |
| 31 | \( 1 - 4.07T + 31T^{2} \) |
| 37 | \( 1 + 7.33T + 37T^{2} \) |
| 41 | \( 1 + 7.61T + 41T^{2} \) |
| 43 | \( 1 - 3.57T + 43T^{2} \) |
| 47 | \( 1 + 0.864T + 47T^{2} \) |
| 53 | \( 1 + 4.84T + 53T^{2} \) |
| 59 | \( 1 - 2.86T + 59T^{2} \) |
| 61 | \( 1 - 8.17T + 61T^{2} \) |
| 67 | \( 1 - 6.80T + 67T^{2} \) |
| 71 | \( 1 + 1.78T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 + 9.39T + 79T^{2} \) |
| 83 | \( 1 - 6.30T + 83T^{2} \) |
| 89 | \( 1 - 3.25T + 89T^{2} \) |
| 97 | \( 1 - 4.97T + 97T^{2} \) |
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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
−7.40937477945706394065052071674, −6.72225307561713136376172996503, −6.20966055721093667138598929283, −5.35777282747358616733203275717, −4.88246907263921079404536023943, −3.89293937498383354211729065769, −3.13254238592372228760152326835, −2.07424154909793939839053541953, −0.951508851710218584946822495313, 0,
0.951508851710218584946822495313, 2.07424154909793939839053541953, 3.13254238592372228760152326835, 3.89293937498383354211729065769, 4.88246907263921079404536023943, 5.35777282747358616733203275717, 6.20966055721093667138598929283, 6.72225307561713136376172996503, 7.40937477945706394065052071674