L(s) = 1 | − 2.68·2-s − 3-s + 5.22·4-s + 2.68·6-s + 1.68·7-s − 8.65·8-s + 9-s + 1.07·11-s − 5.22·12-s + 2.67·13-s − 4.53·14-s + 12.8·16-s − 3.93·17-s − 2.68·18-s − 1.17·19-s − 1.68·21-s − 2.89·22-s − 4.06·23-s + 8.65·24-s − 7.19·26-s − 27-s + 8.80·28-s + 5.95·29-s − 7.10·31-s − 17.1·32-s − 1.07·33-s + 10.5·34-s + ⋯ |
L(s) = 1 | − 1.90·2-s − 0.577·3-s + 2.61·4-s + 1.09·6-s + 0.637·7-s − 3.05·8-s + 0.333·9-s + 0.325·11-s − 1.50·12-s + 0.742·13-s − 1.21·14-s + 3.20·16-s − 0.953·17-s − 0.633·18-s − 0.270·19-s − 0.368·21-s − 0.618·22-s − 0.847·23-s + 1.76·24-s − 1.41·26-s − 0.192·27-s + 1.66·28-s + 1.10·29-s − 1.27·31-s − 3.02·32-s − 0.187·33-s + 1.81·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + 2.68T + 2T^{2} \) |
| 7 | \( 1 - 1.68T + 7T^{2} \) |
| 11 | \( 1 - 1.07T + 11T^{2} \) |
| 13 | \( 1 - 2.67T + 13T^{2} \) |
| 17 | \( 1 + 3.93T + 17T^{2} \) |
| 19 | \( 1 + 1.17T + 19T^{2} \) |
| 23 | \( 1 + 4.06T + 23T^{2} \) |
| 29 | \( 1 - 5.95T + 29T^{2} \) |
| 31 | \( 1 + 7.10T + 31T^{2} \) |
| 37 | \( 1 + 4.58T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 2.58T + 43T^{2} \) |
| 47 | \( 1 + 1.91T + 47T^{2} \) |
| 53 | \( 1 + 2.54T + 53T^{2} \) |
| 59 | \( 1 - 1.33T + 59T^{2} \) |
| 61 | \( 1 + 7.28T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + 5.98T + 71T^{2} \) |
| 73 | \( 1 + 3.30T + 73T^{2} \) |
| 79 | \( 1 + 4.00T + 79T^{2} \) |
| 83 | \( 1 - 8.87T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
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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
−8.748298348507309658513881274966, −8.299918635788798208327651840634, −7.43560564742744717456074152400, −6.62364071826257437210144552415, −6.11336251240664284501718849224, −4.92097941843277901803372304721, −3.58551772316829954831444845551, −2.13483728993346478922184714353, −1.36883888584412572952703562600, 0,
1.36883888584412572952703562600, 2.13483728993346478922184714353, 3.58551772316829954831444845551, 4.92097941843277901803372304721, 6.11336251240664284501718849224, 6.62364071826257437210144552415, 7.43560564742744717456074152400, 8.299918635788798208327651840634, 8.748298348507309658513881274966