L(s) = 1 | + 1.27·3-s + 0.726·7-s − 1.37·9-s + 0.273·11-s + 5.95·13-s − 5.27·17-s − 19-s + 0.924·21-s − 3.67·23-s − 5.57·27-s − 2.27·29-s − 3.19·31-s + 0.348·33-s − 8.12·37-s + 7.58·39-s − 9.43·41-s + 9.81·43-s + 12.1·47-s − 6.47·49-s − 6.71·51-s − 5.69·53-s − 1.27·57-s + 4.20·59-s − 0.103·61-s − 0.999·63-s − 11.7·67-s − 4.68·69-s + ⋯ |
L(s) = 1 | + 0.735·3-s + 0.274·7-s − 0.459·9-s + 0.0825·11-s + 1.65·13-s − 1.27·17-s − 0.229·19-s + 0.201·21-s − 0.767·23-s − 1.07·27-s − 0.422·29-s − 0.574·31-s + 0.0607·33-s − 1.33·37-s + 1.21·39-s − 1.47·41-s + 1.49·43-s + 1.77·47-s − 0.924·49-s − 0.940·51-s − 0.782·53-s − 0.168·57-s + 0.547·59-s − 0.0132·61-s − 0.125·63-s − 1.43·67-s − 0.564·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 - 1.27T + 3T^{2} \) |
| 7 | \( 1 - 0.726T + 7T^{2} \) |
| 11 | \( 1 - 0.273T + 11T^{2} \) |
| 13 | \( 1 - 5.95T + 13T^{2} \) |
| 17 | \( 1 + 5.27T + 17T^{2} \) |
| 23 | \( 1 + 3.67T + 23T^{2} \) |
| 29 | \( 1 + 2.27T + 29T^{2} \) |
| 31 | \( 1 + 3.19T + 31T^{2} \) |
| 37 | \( 1 + 8.12T + 37T^{2} \) |
| 41 | \( 1 + 9.43T + 41T^{2} \) |
| 43 | \( 1 - 9.81T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 5.69T + 53T^{2} \) |
| 59 | \( 1 - 4.20T + 59T^{2} \) |
| 61 | \( 1 + 0.103T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 + 5.75T + 71T^{2} \) |
| 73 | \( 1 + 6.67T + 73T^{2} \) |
| 79 | \( 1 + 3.87T + 79T^{2} \) |
| 83 | \( 1 + 0.488T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 - 4.44T + 97T^{2} \) |
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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
−7.64530390296043936598963081622, −6.89244690367551691333498493110, −6.06403308641084553298425315647, −5.61000459636287331195023958020, −4.49482366776071365596298165351, −3.84035847487999676192514607206, −3.19106099124531897616364711897, −2.20923838849575574211228593640, −1.51671879359981659705205137181, 0,
1.51671879359981659705205137181, 2.20923838849575574211228593640, 3.19106099124531897616364711897, 3.84035847487999676192514607206, 4.49482366776071365596298165351, 5.61000459636287331195023958020, 6.06403308641084553298425315647, 6.89244690367551691333498493110, 7.64530390296043936598963081622