| L(s) = 1 | − 2-s + 0.0257·3-s + 4-s + 3.06·5-s − 0.0257·6-s − 7-s − 8-s − 2.99·9-s − 3.06·10-s − 2.07·11-s + 0.0257·12-s + 1.32·13-s + 14-s + 0.0790·15-s + 16-s − 1.76·17-s + 2.99·18-s + 6.17·19-s + 3.06·20-s − 0.0257·21-s + 2.07·22-s − 8.31·23-s − 0.0257·24-s + 4.41·25-s − 1.32·26-s − 0.154·27-s − 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.0148·3-s + 0.5·4-s + 1.37·5-s − 0.0105·6-s − 0.377·7-s − 0.353·8-s − 0.999·9-s − 0.970·10-s − 0.625·11-s + 0.00743·12-s + 0.367·13-s + 0.267·14-s + 0.0204·15-s + 0.250·16-s − 0.428·17-s + 0.706·18-s + 1.41·19-s + 0.685·20-s − 0.00562·21-s + 0.442·22-s − 1.73·23-s − 0.00526·24-s + 0.882·25-s − 0.259·26-s − 0.0297·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 - 0.0257T + 3T^{2} \) |
| 5 | \( 1 - 3.06T + 5T^{2} \) |
| 11 | \( 1 + 2.07T + 11T^{2} \) |
| 13 | \( 1 - 1.32T + 13T^{2} \) |
| 17 | \( 1 + 1.76T + 17T^{2} \) |
| 19 | \( 1 - 6.17T + 19T^{2} \) |
| 23 | \( 1 + 8.31T + 23T^{2} \) |
| 29 | \( 1 - 3.32T + 29T^{2} \) |
| 31 | \( 1 + 5.80T + 31T^{2} \) |
| 37 | \( 1 - 8.74T + 37T^{2} \) |
| 41 | \( 1 + 2.56T + 41T^{2} \) |
| 43 | \( 1 + 2.36T + 43T^{2} \) |
| 47 | \( 1 + 5.87T + 47T^{2} \) |
| 53 | \( 1 - 1.59T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 - 5.45T + 61T^{2} \) |
| 67 | \( 1 - 2.72T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 + 8.49T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 - 9.65T + 97T^{2} \) |
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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
−7.929704806627791695638623297728, −6.97551973578233663141960521015, −6.25517268744653808693490060297, −5.67657729715996965549080670885, −5.24623635583102461907046363069, −3.87444811145958172967896138878, −2.84915368735757939510920096729, −2.32462594337591155370255872033, −1.33864702356333953365601419254, 0,
1.33864702356333953365601419254, 2.32462594337591155370255872033, 2.84915368735757939510920096729, 3.87444811145958172967896138878, 5.24623635583102461907046363069, 5.67657729715996965549080670885, 6.25517268744653808693490060297, 6.97551973578233663141960521015, 7.929704806627791695638623297728
