L(s) = 1 | − 2.53·2-s − 3-s + 4.44·4-s − 3.08·5-s + 2.53·6-s − 3.86·7-s − 6.19·8-s + 9-s + 7.83·10-s + 5.14·11-s − 4.44·12-s + 0.926·13-s + 9.80·14-s + 3.08·15-s + 6.83·16-s + 17-s − 2.53·18-s − 2.24·19-s − 13.7·20-s + 3.86·21-s − 13.0·22-s + 1.13·23-s + 6.19·24-s + 4.52·25-s − 2.35·26-s − 27-s − 17.1·28-s + ⋯ |
L(s) = 1 | − 1.79·2-s − 0.577·3-s + 2.22·4-s − 1.38·5-s + 1.03·6-s − 1.46·7-s − 2.18·8-s + 0.333·9-s + 2.47·10-s + 1.55·11-s − 1.28·12-s + 0.256·13-s + 2.62·14-s + 0.796·15-s + 1.70·16-s + 0.242·17-s − 0.598·18-s − 0.514·19-s − 3.06·20-s + 0.843·21-s − 2.78·22-s + 0.237·23-s + 1.26·24-s + 0.905·25-s − 0.461·26-s − 0.192·27-s − 3.24·28-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 | \( 1 + 2.53T + 2T^{2} \) |
| 5 | \( 1 + 3.08T + 5T^{2} \) |
| 7 | \( 1 + 3.86T + 7T^{2} \) |
| 11 | \( 1 - 5.14T + 11T^{2} \) |
| 13 | \( 1 - 0.926T + 13T^{2} \) |
| 19 | \( 1 + 2.24T + 19T^{2} \) |
| 23 | \( 1 - 1.13T + 23T^{2} \) |
| 29 | \( 1 + 3.12T + 29T^{2} \) |
| 31 | \( 1 + 0.0729T + 31T^{2} \) |
| 37 | \( 1 - 0.866T + 37T^{2} \) |
| 41 | \( 1 + 4.08T + 41T^{2} \) |
| 43 | \( 1 + 4.52T + 43T^{2} \) |
| 47 | \( 1 - 2.26T + 47T^{2} \) |
| 53 | \( 1 + 6.59T + 53T^{2} \) |
| 59 | \( 1 - 3.60T + 59T^{2} \) |
| 61 | \( 1 + 2.20T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 - 6.03T + 71T^{2} \) |
| 73 | \( 1 + 8.31T + 73T^{2} \) |
| 79 | \( 1 - 3.23T + 79T^{2} \) |
| 83 | \( 1 - 7.29T + 83T^{2} \) |
| 89 | \( 1 - 5.19T + 89T^{2} \) |
| 97 | \( 1 + 3.45T + 97T^{2} \) |
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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
−7.51021956645951000356555740706, −6.92493796113398623179618830816, −6.49391532999120986765672160572, −5.90604766591498658955701124195, −4.48572118886301702855822171643, −3.66210565549902632263690608913, −3.12111781975594732784561659923, −1.76391787419309141661682533115, −0.77203716587009411509189555222, 0,
0.77203716587009411509189555222, 1.76391787419309141661682533115, 3.12111781975594732784561659923, 3.66210565549902632263690608913, 4.48572118886301702855822171643, 5.90604766591498658955701124195, 6.49391532999120986765672160572, 6.92493796113398623179618830816, 7.51021956645951000356555740706