L(s) = 1 | + 3-s − 5-s − 2.23·7-s + 9-s − 1.61·11-s − 4.23·13-s − 15-s + 1.85·17-s + 5.38·19-s − 2.23·21-s + 1.47·23-s + 25-s + 27-s − 2.14·29-s + 9.47·31-s − 1.61·33-s + 2.23·35-s − 37-s − 4.23·39-s − 7.47·41-s + 7.56·43-s − 45-s − 3.76·47-s − 1.99·49-s + 1.85·51-s + 6.47·53-s + 1.61·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.845·7-s + 0.333·9-s − 0.487·11-s − 1.17·13-s − 0.258·15-s + 0.449·17-s + 1.23·19-s − 0.487·21-s + 0.306·23-s + 0.200·25-s + 0.192·27-s − 0.398·29-s + 1.70·31-s − 0.281·33-s + 0.377·35-s − 0.164·37-s − 0.678·39-s − 1.16·41-s + 1.15·43-s − 0.149·45-s − 0.549·47-s − 0.285·49-s + 0.259·51-s + 0.889·53-s + 0.218·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
good | 7 | \( 1 + 2.23T + 7T^{2} \) |
| 11 | \( 1 + 1.61T + 11T^{2} \) |
| 13 | \( 1 + 4.23T + 13T^{2} \) |
| 17 | \( 1 - 1.85T + 17T^{2} \) |
| 19 | \( 1 - 5.38T + 19T^{2} \) |
| 23 | \( 1 - 1.47T + 23T^{2} \) |
| 29 | \( 1 + 2.14T + 29T^{2} \) |
| 31 | \( 1 - 9.47T + 31T^{2} \) |
| 41 | \( 1 + 7.47T + 41T^{2} \) |
| 43 | \( 1 - 7.56T + 43T^{2} \) |
| 47 | \( 1 + 3.76T + 47T^{2} \) |
| 53 | \( 1 - 6.47T + 53T^{2} \) |
| 59 | \( 1 - 5.32T + 59T^{2} \) |
| 61 | \( 1 + 4.52T + 61T^{2} \) |
| 67 | \( 1 + 6.32T + 67T^{2} \) |
| 71 | \( 1 + 8.61T + 71T^{2} \) |
| 73 | \( 1 + 6.56T + 73T^{2} \) |
| 79 | \( 1 - 5.61T + 79T^{2} \) |
| 83 | \( 1 + 4.56T + 83T^{2} \) |
| 89 | \( 1 + 0.0557T + 89T^{2} \) |
| 97 | \( 1 - 7.47T + 97T^{2} \) |
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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
−7.43705541229209344991701924692, −6.95043320648695941432578012154, −6.08366669319899678333206092847, −5.21961616848257865209397030731, −4.62631091220408083192307008943, −3.67871929423913585999187962736, −3.00073924263771598401952325747, −2.51083281596909422925206822557, −1.19393180138955967855452823620, 0,
1.19393180138955967855452823620, 2.51083281596909422925206822557, 3.00073924263771598401952325747, 3.67871929423913585999187962736, 4.62631091220408083192307008943, 5.21961616848257865209397030731, 6.08366669319899678333206092847, 6.95043320648695941432578012154, 7.43705541229209344991701924692