L(s) = 1 | + 2.02·2-s − 3-s + 2.08·4-s + 4.24·5-s − 2.02·6-s − 1.21·7-s + 0.167·8-s + 9-s + 8.57·10-s + 4.05·11-s − 2.08·12-s + 13-s − 2.45·14-s − 4.24·15-s − 3.82·16-s − 3.43·17-s + 2.02·18-s + 1.65·19-s + 8.83·20-s + 1.21·21-s + 8.20·22-s + 0.620·23-s − 0.167·24-s + 13.0·25-s + 2.02·26-s − 27-s − 2.53·28-s + ⋯ |
L(s) = 1 | + 1.42·2-s − 0.577·3-s + 1.04·4-s + 1.89·5-s − 0.824·6-s − 0.459·7-s + 0.0591·8-s + 0.333·9-s + 2.71·10-s + 1.22·11-s − 0.601·12-s + 0.277·13-s − 0.656·14-s − 1.09·15-s − 0.956·16-s − 0.832·17-s + 0.476·18-s + 0.380·19-s + 1.97·20-s + 0.265·21-s + 1.74·22-s + 0.129·23-s − 0.0341·24-s + 2.60·25-s + 0.396·26-s − 0.192·27-s − 0.478·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.055868117\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.055868117\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
good | 2 | \( 1 - 2.02T + 2T^{2} \) |
| 5 | \( 1 - 4.24T + 5T^{2} \) |
| 7 | \( 1 + 1.21T + 7T^{2} \) |
| 11 | \( 1 - 4.05T + 11T^{2} \) |
| 17 | \( 1 + 3.43T + 17T^{2} \) |
| 19 | \( 1 - 1.65T + 19T^{2} \) |
| 23 | \( 1 - 0.620T + 23T^{2} \) |
| 29 | \( 1 - 7.32T + 29T^{2} \) |
| 31 | \( 1 - 3.81T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 + 1.65T + 41T^{2} \) |
| 43 | \( 1 - 9.74T + 43T^{2} \) |
| 47 | \( 1 - 9.90T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 + 7.24T + 59T^{2} \) |
| 61 | \( 1 + 5.43T + 61T^{2} \) |
| 67 | \( 1 - 0.936T + 67T^{2} \) |
| 71 | \( 1 - 2.07T + 71T^{2} \) |
| 73 | \( 1 + 4.79T + 73T^{2} \) |
| 79 | \( 1 + 5.55T + 79T^{2} \) |
| 83 | \( 1 - 6.01T + 83T^{2} \) |
| 89 | \( 1 + 3.23T + 89T^{2} \) |
| 97 | \( 1 - 17.3T + 97T^{2} \) |
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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
−8.906760944600819187097221312152, −7.06589966745887002361571138913, −6.52922179097748087621216468713, −6.14139296413683463622621083883, −5.51427511691591245154498638990, −4.80864270998196810099274774336, −4.05121785644514529604388704712, −3.02052095444912993998029350093, −2.20860894071394172637907869465, −1.16230956297795650499637477637,
1.16230956297795650499637477637, 2.20860894071394172637907869465, 3.02052095444912993998029350093, 4.05121785644514529604388704712, 4.80864270998196810099274774336, 5.51427511691591245154498638990, 6.14139296413683463622621083883, 6.52922179097748087621216468713, 7.06589966745887002361571138913, 8.906760944600819187097221312152