L(s) = 1 | − 2.14·2-s + 2.61·4-s + 1.42·5-s + 3.61·7-s − 1.32·8-s − 3.06·10-s − 4.28·11-s + 0.939·13-s − 7.77·14-s − 2.38·16-s − 7.48·17-s + 4.97·19-s + 3.72·20-s + 9.21·22-s − 23-s − 2.97·25-s − 2.01·26-s + 9.46·28-s + 29-s − 2.22·31-s + 7.77·32-s + 16.0·34-s + 5.15·35-s − 0.191·37-s − 10.6·38-s − 1.88·40-s + 5.36·41-s + ⋯ |
L(s) = 1 | − 1.51·2-s + 1.30·4-s + 0.637·5-s + 1.36·7-s − 0.467·8-s − 0.967·10-s − 1.29·11-s + 0.260·13-s − 2.07·14-s − 0.597·16-s − 1.81·17-s + 1.14·19-s + 0.833·20-s + 1.96·22-s − 0.208·23-s − 0.594·25-s − 0.395·26-s + 1.78·28-s + 0.185·29-s − 0.400·31-s + 1.37·32-s + 2.75·34-s + 0.870·35-s − 0.0314·37-s − 1.73·38-s − 0.297·40-s + 0.838·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 + 2.14T + 2T^{2} \) |
| 5 | \( 1 - 1.42T + 5T^{2} \) |
| 7 | \( 1 - 3.61T + 7T^{2} \) |
| 11 | \( 1 + 4.28T + 11T^{2} \) |
| 13 | \( 1 - 0.939T + 13T^{2} \) |
| 17 | \( 1 + 7.48T + 17T^{2} \) |
| 19 | \( 1 - 4.97T + 19T^{2} \) |
| 31 | \( 1 + 2.22T + 31T^{2} \) |
| 37 | \( 1 + 0.191T + 37T^{2} \) |
| 41 | \( 1 - 5.36T + 41T^{2} \) |
| 43 | \( 1 + 6.26T + 43T^{2} \) |
| 47 | \( 1 + 0.471T + 47T^{2} \) |
| 53 | \( 1 - 6.09T + 53T^{2} \) |
| 59 | \( 1 + 7.35T + 59T^{2} \) |
| 61 | \( 1 - 9.04T + 61T^{2} \) |
| 67 | \( 1 - 8.32T + 67T^{2} \) |
| 71 | \( 1 + 4.71T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 + 7.80T + 79T^{2} \) |
| 83 | \( 1 - 16.6T + 83T^{2} \) |
| 89 | \( 1 + 2.37T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.933616911195541320172068968639, −7.33623874035334252516608459086, −6.59014829703700167379952083497, −5.58673944731189459273472078300, −4.98966882701864219683756256271, −4.16073601451577092905839133468, −2.64508538563103299275226894254, −2.04894148620891877358507456327, −1.28841267417587618007365975830, 0,
1.28841267417587618007365975830, 2.04894148620891877358507456327, 2.64508538563103299275226894254, 4.16073601451577092905839133468, 4.98966882701864219683756256271, 5.58673944731189459273472078300, 6.59014829703700167379952083497, 7.33623874035334252516608459086, 7.933616911195541320172068968639