L(s) = 1 | + 1.11·2-s − 2.87·3-s − 0.760·4-s − 3.20·6-s + 7-s − 3.07·8-s + 5.29·9-s − 1.44·11-s + 2.18·12-s + 2.99·13-s + 1.11·14-s − 1.90·16-s + 0.663·17-s + 5.89·18-s − 6.91·19-s − 2.87·21-s − 1.61·22-s − 23-s + 8.85·24-s + 3.33·26-s − 6.60·27-s − 0.760·28-s + 2.85·29-s + 5.80·31-s + 4.02·32-s + 4.17·33-s + 0.738·34-s + ⋯ |
L(s) = 1 | + 0.787·2-s − 1.66·3-s − 0.380·4-s − 1.30·6-s + 0.377·7-s − 1.08·8-s + 1.76·9-s − 0.436·11-s + 0.632·12-s + 0.830·13-s + 0.297·14-s − 0.475·16-s + 0.160·17-s + 1.38·18-s − 1.58·19-s − 0.628·21-s − 0.343·22-s − 0.208·23-s + 1.80·24-s + 0.653·26-s − 1.27·27-s − 0.143·28-s + 0.529·29-s + 1.04·31-s + 0.712·32-s + 0.726·33-s + 0.126·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 - 1.11T + 2T^{2} \) |
| 3 | \( 1 + 2.87T + 3T^{2} \) |
| 11 | \( 1 + 1.44T + 11T^{2} \) |
| 13 | \( 1 - 2.99T + 13T^{2} \) |
| 17 | \( 1 - 0.663T + 17T^{2} \) |
| 19 | \( 1 + 6.91T + 19T^{2} \) |
| 29 | \( 1 - 2.85T + 29T^{2} \) |
| 31 | \( 1 - 5.80T + 31T^{2} \) |
| 37 | \( 1 - 1.45T + 37T^{2} \) |
| 41 | \( 1 - 1.31T + 41T^{2} \) |
| 43 | \( 1 + 6.04T + 43T^{2} \) |
| 47 | \( 1 - 7.26T + 47T^{2} \) |
| 53 | \( 1 - 7.52T + 53T^{2} \) |
| 59 | \( 1 + 1.21T + 59T^{2} \) |
| 61 | \( 1 - 1.29T + 61T^{2} \) |
| 67 | \( 1 - 6.72T + 67T^{2} \) |
| 71 | \( 1 + 15.4T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 + 17.4T + 89T^{2} \) |
| 97 | \( 1 + 2.17T + 97T^{2} \) |
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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
−8.151504074419436686144746474567, −6.92692267278085484597988118485, −6.29155441213271824453281596836, −5.77057836936723920183994708837, −5.11159351299474909169452613555, −4.42503584184753423054651327395, −3.89953618272354020287221119404, −2.56858064795828530634241904022, −1.13062448652891500286785544467, 0,
1.13062448652891500286785544467, 2.56858064795828530634241904022, 3.89953618272354020287221119404, 4.42503584184753423054651327395, 5.11159351299474909169452613555, 5.77057836936723920183994708837, 6.29155441213271824453281596836, 6.92692267278085484597988118485, 8.151504074419436686144746474567