L(s) = 1 | + 0.596·2-s − 3.22·3-s − 1.64·4-s − 5-s − 1.92·6-s + 0.0914·7-s − 2.17·8-s + 7.38·9-s − 0.596·10-s + 0.938·11-s + 5.29·12-s + 6.40·13-s + 0.0545·14-s + 3.22·15-s + 1.99·16-s − 5.47·17-s + 4.40·18-s − 4.62·19-s + 1.64·20-s − 0.294·21-s + 0.559·22-s + 9.45·23-s + 7.00·24-s + 25-s + 3.82·26-s − 14.1·27-s − 0.150·28-s + ⋯ |
L(s) = 1 | + 0.421·2-s − 1.86·3-s − 0.821·4-s − 0.447·5-s − 0.784·6-s + 0.0345·7-s − 0.768·8-s + 2.46·9-s − 0.188·10-s + 0.282·11-s + 1.52·12-s + 1.77·13-s + 0.0145·14-s + 0.831·15-s + 0.497·16-s − 1.32·17-s + 1.03·18-s − 1.06·19-s + 0.367·20-s − 0.0642·21-s + 0.119·22-s + 1.97·23-s + 1.43·24-s + 0.200·25-s + 0.750·26-s − 2.71·27-s − 0.0284·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 241 | \( 1 + T \) |
good | 2 | \( 1 - 0.596T + 2T^{2} \) |
| 3 | \( 1 + 3.22T + 3T^{2} \) |
| 7 | \( 1 - 0.0914T + 7T^{2} \) |
| 11 | \( 1 - 0.938T + 11T^{2} \) |
| 13 | \( 1 - 6.40T + 13T^{2} \) |
| 17 | \( 1 + 5.47T + 17T^{2} \) |
| 19 | \( 1 + 4.62T + 19T^{2} \) |
| 23 | \( 1 - 9.45T + 23T^{2} \) |
| 29 | \( 1 + 0.666T + 29T^{2} \) |
| 31 | \( 1 + 7.51T + 31T^{2} \) |
| 37 | \( 1 - 5.97T + 37T^{2} \) |
| 41 | \( 1 + 1.82T + 41T^{2} \) |
| 43 | \( 1 - 2.41T + 43T^{2} \) |
| 47 | \( 1 - 1.50T + 47T^{2} \) |
| 53 | \( 1 + 3.52T + 53T^{2} \) |
| 59 | \( 1 + 14.3T + 59T^{2} \) |
| 61 | \( 1 + 6.52T + 61T^{2} \) |
| 67 | \( 1 + 15.3T + 67T^{2} \) |
| 71 | \( 1 - 6.49T + 71T^{2} \) |
| 73 | \( 1 + 2.73T + 73T^{2} \) |
| 79 | \( 1 + 8.80T + 79T^{2} \) |
| 83 | \( 1 + 0.402T + 83T^{2} \) |
| 89 | \( 1 - 17.9T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−9.222387101338677456666593176935, −8.759512757141960087518696391694, −7.44379330577430721311675529429, −6.34090005748244908807658859040, −6.08892067753312352914600245953, −4.91044843334447599623148932805, −4.43461418716682980305604622870, −3.53947405358228233209793816478, −1.27556383279873247273191131019, 0,
1.27556383279873247273191131019, 3.53947405358228233209793816478, 4.43461418716682980305604622870, 4.91044843334447599623148932805, 6.08892067753312352914600245953, 6.34090005748244908807658859040, 7.44379330577430721311675529429, 8.759512757141960087518696391694, 9.222387101338677456666593176935