L(s) = 1 | − 2.30·2-s − 3-s + 3.30·4-s + 2.30·6-s − 7-s − 3.00·8-s + 9-s − 11-s − 3.30·12-s − 1.30·13-s + 2.30·14-s + 0.302·16-s − 4.60·17-s − 2.30·18-s − 6.30·19-s + 21-s + 2.30·22-s − 1.60·23-s + 3.00·24-s + 3·26-s − 27-s − 3.30·28-s + 9.21·29-s + 9.60·31-s + 5.30·32-s + 33-s + 10.6·34-s + ⋯ |
L(s) = 1 | − 1.62·2-s − 0.577·3-s + 1.65·4-s + 0.940·6-s − 0.377·7-s − 1.06·8-s + 0.333·9-s − 0.301·11-s − 0.953·12-s − 0.361·13-s + 0.615·14-s + 0.0756·16-s − 1.11·17-s − 0.542·18-s − 1.44·19-s + 0.218·21-s + 0.490·22-s − 0.334·23-s + 0.612·24-s + 0.588·26-s − 0.192·27-s − 0.624·28-s + 1.71·29-s + 1.72·31-s + 0.937·32-s + 0.174·33-s + 1.81·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 13 | \( 1 + 1.30T + 13T^{2} \) |
| 17 | \( 1 + 4.60T + 17T^{2} \) |
| 19 | \( 1 + 6.30T + 19T^{2} \) |
| 23 | \( 1 + 1.60T + 23T^{2} \) |
| 29 | \( 1 - 9.21T + 29T^{2} \) |
| 31 | \( 1 - 9.60T + 31T^{2} \) |
| 37 | \( 1 + 7.51T + 37T^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 - 7.21T + 43T^{2} \) |
| 47 | \( 1 - 6.21T + 47T^{2} \) |
| 53 | \( 1 + 4.39T + 53T^{2} \) |
| 59 | \( 1 - 2.30T + 59T^{2} \) |
| 61 | \( 1 + 2.39T + 61T^{2} \) |
| 67 | \( 1 + 6.60T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 1.78T + 73T^{2} \) |
| 79 | \( 1 - 11T + 79T^{2} \) |
| 83 | \( 1 - 8.30T + 83T^{2} \) |
| 89 | \( 1 + 9T + 89T^{2} \) |
| 97 | \( 1 + 4.30T + 97T^{2} \) |
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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
−7.85574363226124867204286252651, −7.17938115583077778200203482765, −6.41169136752711370773103317282, −6.13772083490541776991439732456, −4.76917546656680670560344310292, −4.24999880457103174627469133965, −2.73969082149708782507516451624, −2.14146356286875620326466302853, −0.919137424467534711701947506833, 0,
0.919137424467534711701947506833, 2.14146356286875620326466302853, 2.73969082149708782507516451624, 4.24999880457103174627469133965, 4.76917546656680670560344310292, 6.13772083490541776991439732456, 6.41169136752711370773103317282, 7.17938115583077778200203482765, 7.85574363226124867204286252651