L(s) = 1 | − 2-s + 4-s + 3.44·5-s + 2·7-s − 8-s − 3.44·10-s + 1.44·11-s + 2.89·13-s − 2·14-s + 16-s − 2·17-s − 2·19-s + 3.44·20-s − 1.44·22-s + 23-s + 6.89·25-s − 2.89·26-s + 2·28-s + 0.898·29-s − 1.89·31-s − 32-s + 2·34-s + 6.89·35-s + 2.55·37-s + 2·38-s − 3.44·40-s + 1.89·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.54·5-s + 0.755·7-s − 0.353·8-s − 1.09·10-s + 0.437·11-s + 0.804·13-s − 0.534·14-s + 0.250·16-s − 0.485·17-s − 0.458·19-s + 0.771·20-s − 0.309·22-s + 0.208·23-s + 1.37·25-s − 0.568·26-s + 0.377·28-s + 0.166·29-s − 0.341·31-s − 0.176·32-s + 0.342·34-s + 1.16·35-s + 0.419·37-s + 0.324·38-s − 0.545·40-s + 0.296·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.246233968\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.246233968\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 - 3.44T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 1.44T + 11T^{2} \) |
| 13 | \( 1 - 2.89T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 29 | \( 1 - 0.898T + 29T^{2} \) |
| 31 | \( 1 + 1.89T + 31T^{2} \) |
| 37 | \( 1 - 2.55T + 37T^{2} \) |
| 41 | \( 1 - 1.89T + 41T^{2} \) |
| 43 | \( 1 - 10T + 43T^{2} \) |
| 47 | \( 1 + 9.79T + 47T^{2} \) |
| 53 | \( 1 - 7.44T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 15.4T + 61T^{2} \) |
| 67 | \( 1 - 6.34T + 67T^{2} \) |
| 71 | \( 1 - 9T + 71T^{2} \) |
| 73 | \( 1 + 3.79T + 73T^{2} \) |
| 79 | \( 1 - 1.10T + 79T^{2} \) |
| 83 | \( 1 + 7.24T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 9.79T + 97T^{2} \) |
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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
−8.649030123689174404263380735519, −7.982644673241960545199345908097, −6.95185741724099914931687183108, −6.33585645416526007419823447525, −5.71899595241238865361530114877, −4.89785870900565687855448652396, −3.84625362352221746424992358001, −2.56677764867380324246447465545, −1.86302673409034955735711793534, −1.05034484626692785858240134261,
1.05034484626692785858240134261, 1.86302673409034955735711793534, 2.56677764867380324246447465545, 3.84625362352221746424992358001, 4.89785870900565687855448652396, 5.71899595241238865361530114877, 6.33585645416526007419823447525, 6.95185741724099914931687183108, 7.982644673241960545199345908097, 8.649030123689174404263380735519