L(s) = 1 | + 2.23·2-s + 3.00·4-s + 2·5-s + 7-s + 2.23·8-s + 4.47·10-s + 11-s − 1.23·13-s + 2.23·14-s − 0.999·16-s − 1.23·17-s − 2.47·19-s + 6.00·20-s + 2.23·22-s + 6.47·23-s − 25-s − 2.76·26-s + 3.00·28-s + 0.472·29-s − 7.23·31-s − 6.70·32-s − 2.76·34-s + 2·35-s + 0.472·37-s − 5.52·38-s + 4.47·40-s + 6.76·41-s + ⋯ |
L(s) = 1 | + 1.58·2-s + 1.50·4-s + 0.894·5-s + 0.377·7-s + 0.790·8-s + 1.41·10-s + 0.301·11-s − 0.342·13-s + 0.597·14-s − 0.249·16-s − 0.299·17-s − 0.567·19-s + 1.34·20-s + 0.476·22-s + 1.34·23-s − 0.200·25-s − 0.542·26-s + 0.566·28-s + 0.0876·29-s − 1.29·31-s − 1.18·32-s − 0.474·34-s + 0.338·35-s + 0.0776·37-s − 0.896·38-s + 0.707·40-s + 1.05·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.016885152\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.016885152\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 - 2.23T + 2T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 17 | \( 1 + 1.23T + 17T^{2} \) |
| 19 | \( 1 + 2.47T + 19T^{2} \) |
| 23 | \( 1 - 6.47T + 23T^{2} \) |
| 29 | \( 1 - 0.472T + 29T^{2} \) |
| 31 | \( 1 + 7.23T + 31T^{2} \) |
| 37 | \( 1 - 0.472T + 37T^{2} \) |
| 41 | \( 1 - 6.76T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 7.23T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 + 3.23T + 59T^{2} \) |
| 61 | \( 1 + 2.76T + 61T^{2} \) |
| 67 | \( 1 - 5.52T + 67T^{2} \) |
| 71 | \( 1 - 1.52T + 71T^{2} \) |
| 73 | \( 1 + 5.23T + 73T^{2} \) |
| 79 | \( 1 - 8.94T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 9.41T + 97T^{2} \) |
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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
−10.88226261751898714785691828972, −9.618661278723839710935866831226, −8.875903203689607430711580846881, −7.48512361075406458516301979045, −6.55903344230073039738978596285, −5.78469681677328770050691654018, −4.99203528635377175857833357849, −4.12946067720582032634556591522, −2.90741800162899026863915305070, −1.86016951804562747704185740948,
1.86016951804562747704185740948, 2.90741800162899026863915305070, 4.12946067720582032634556591522, 4.99203528635377175857833357849, 5.78469681677328770050691654018, 6.55903344230073039738978596285, 7.48512361075406458516301979045, 8.875903203689607430711580846881, 9.618661278723839710935866831226, 10.88226261751898714785691828972