L(s) = 1 | − 1.97·3-s − 5-s − 2.05·7-s + 0.905·9-s + 1.76·11-s − 3.78·13-s + 1.97·15-s + 2.63·17-s − 1.86·19-s + 4.06·21-s − 0.710·23-s + 25-s + 4.13·27-s + 5.21·29-s + 1.26·31-s − 3.47·33-s + 2.05·35-s − 9.36·37-s + 7.47·39-s − 1.82·41-s + 5.93·43-s − 0.905·45-s + 2.90·47-s − 2.76·49-s − 5.20·51-s + 4.48·53-s − 1.76·55-s + ⋯ |
L(s) = 1 | − 1.14·3-s − 0.447·5-s − 0.778·7-s + 0.301·9-s + 0.530·11-s − 1.04·13-s + 0.510·15-s + 0.638·17-s − 0.427·19-s + 0.887·21-s − 0.148·23-s + 0.200·25-s + 0.796·27-s + 0.968·29-s + 0.227·31-s − 0.605·33-s + 0.347·35-s − 1.54·37-s + 1.19·39-s − 0.285·41-s + 0.905·43-s − 0.134·45-s + 0.423·47-s − 0.394·49-s − 0.728·51-s + 0.615·53-s − 0.237·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 151 | \( 1 - T \) |
good | 3 | \( 1 + 1.97T + 3T^{2} \) |
| 7 | \( 1 + 2.05T + 7T^{2} \) |
| 11 | \( 1 - 1.76T + 11T^{2} \) |
| 13 | \( 1 + 3.78T + 13T^{2} \) |
| 17 | \( 1 - 2.63T + 17T^{2} \) |
| 19 | \( 1 + 1.86T + 19T^{2} \) |
| 23 | \( 1 + 0.710T + 23T^{2} \) |
| 29 | \( 1 - 5.21T + 29T^{2} \) |
| 31 | \( 1 - 1.26T + 31T^{2} \) |
| 37 | \( 1 + 9.36T + 37T^{2} \) |
| 41 | \( 1 + 1.82T + 41T^{2} \) |
| 43 | \( 1 - 5.93T + 43T^{2} \) |
| 47 | \( 1 - 2.90T + 47T^{2} \) |
| 53 | \( 1 - 4.48T + 53T^{2} \) |
| 59 | \( 1 - 9.37T + 59T^{2} \) |
| 61 | \( 1 - 4.23T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 - 3.96T + 71T^{2} \) |
| 73 | \( 1 - 9.95T + 73T^{2} \) |
| 79 | \( 1 - 11.5T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 + 11.7T + 89T^{2} \) |
| 97 | \( 1 + 18.5T + 97T^{2} \) |
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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
−7.56655676732459308733102374979, −6.76352283948020587363645152994, −6.45091488470722231286246456478, −5.52266058456136997536794784283, −4.99241643537668114334084359588, −4.12706512581038861828866826963, −3.31086082882210484980626726931, −2.36564932272200576620701826541, −0.947060117721956217451121783845, 0,
0.947060117721956217451121783845, 2.36564932272200576620701826541, 3.31086082882210484980626726931, 4.12706512581038861828866826963, 4.99241643537668114334084359588, 5.52266058456136997536794784283, 6.45091488470722231286246456478, 6.76352283948020587363645152994, 7.56655676732459308733102374979