| L(s) = 1 | − 2-s − 2.18·3-s + 4-s − 3.08·5-s + 2.18·6-s − 8-s + 1.76·9-s + 3.08·10-s + 1.85·11-s − 2.18·12-s − 13-s + 6.74·15-s + 16-s + 6.50·17-s − 1.76·18-s − 4.58·19-s − 3.08·20-s − 1.85·22-s + 1.02·23-s + 2.18·24-s + 4.53·25-s + 26-s + 2.68·27-s + 8.74·29-s − 6.74·30-s − 0.141·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.26·3-s + 0.5·4-s − 1.38·5-s + 0.891·6-s − 0.353·8-s + 0.589·9-s + 0.976·10-s + 0.560·11-s − 0.630·12-s − 0.277·13-s + 1.74·15-s + 0.250·16-s + 1.57·17-s − 0.417·18-s − 1.05·19-s − 0.690·20-s − 0.396·22-s + 0.214·23-s + 0.445·24-s + 0.907·25-s + 0.196·26-s + 0.517·27-s + 1.62·29-s − 1.23·30-s − 0.0254·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 2.18T + 3T^{2} \) |
| 5 | \( 1 + 3.08T + 5T^{2} \) |
| 11 | \( 1 - 1.85T + 11T^{2} \) |
| 17 | \( 1 - 6.50T + 17T^{2} \) |
| 19 | \( 1 + 4.58T + 19T^{2} \) |
| 23 | \( 1 - 1.02T + 23T^{2} \) |
| 29 | \( 1 - 8.74T + 29T^{2} \) |
| 31 | \( 1 + 0.141T + 31T^{2} \) |
| 37 | \( 1 + 3.23T + 37T^{2} \) |
| 41 | \( 1 + 9.64T + 41T^{2} \) |
| 43 | \( 1 - 4.10T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 + 2.43T + 53T^{2} \) |
| 59 | \( 1 + 9.84T + 59T^{2} \) |
| 61 | \( 1 + 8.11T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 + 8.65T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 + 0.877T + 79T^{2} \) |
| 83 | \( 1 - 8.78T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 5.79T + 97T^{2} \) |
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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
−9.241347091833820383503337667555, −8.349277186696702791199480357944, −7.66962349741765397773451081210, −6.83105078915394701776498028848, −6.11146469008840438636200979954, −5.06733165439942148269517615154, −4.14450600863751700502633691362, −3.04010179483959380917642936772, −1.16138966269533442196314737088, 0,
1.16138966269533442196314737088, 3.04010179483959380917642936772, 4.14450600863751700502633691362, 5.06733165439942148269517615154, 6.11146469008840438636200979954, 6.83105078915394701776498028848, 7.66962349741765397773451081210, 8.349277186696702791199480357944, 9.241347091833820383503337667555
