| L(s) = 1 | + 0.308·3-s − 3.30·5-s − 2.90·9-s + 1.51·11-s + 1.10·13-s − 1.01·15-s + 1.60·17-s + 0.880·19-s + 0.682·23-s + 5.89·25-s − 1.81·27-s − 1.88·29-s + 4.71·31-s + 0.467·33-s − 3.22·37-s + 0.341·39-s + 41-s − 9.57·43-s + 9.59·45-s + 4.78·47-s + 0.495·51-s − 1.63·53-s − 5.00·55-s + 0.271·57-s + 1.01·59-s − 0.380·61-s − 3.66·65-s + ⋯ |
| L(s) = 1 | + 0.177·3-s − 1.47·5-s − 0.968·9-s + 0.457·11-s + 0.307·13-s − 0.262·15-s + 0.389·17-s + 0.202·19-s + 0.142·23-s + 1.17·25-s − 0.350·27-s − 0.349·29-s + 0.846·31-s + 0.0813·33-s − 0.529·37-s + 0.0547·39-s + 0.156·41-s − 1.45·43-s + 1.42·45-s + 0.698·47-s + 0.0693·51-s − 0.224·53-s − 0.675·55-s + 0.0359·57-s + 0.131·59-s − 0.0487·61-s − 0.454·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 - 0.308T + 3T^{2} \) |
| 5 | \( 1 + 3.30T + 5T^{2} \) |
| 11 | \( 1 - 1.51T + 11T^{2} \) |
| 13 | \( 1 - 1.10T + 13T^{2} \) |
| 17 | \( 1 - 1.60T + 17T^{2} \) |
| 19 | \( 1 - 0.880T + 19T^{2} \) |
| 23 | \( 1 - 0.682T + 23T^{2} \) |
| 29 | \( 1 + 1.88T + 29T^{2} \) |
| 31 | \( 1 - 4.71T + 31T^{2} \) |
| 37 | \( 1 + 3.22T + 37T^{2} \) |
| 43 | \( 1 + 9.57T + 43T^{2} \) |
| 47 | \( 1 - 4.78T + 47T^{2} \) |
| 53 | \( 1 + 1.63T + 53T^{2} \) |
| 59 | \( 1 - 1.01T + 59T^{2} \) |
| 61 | \( 1 + 0.380T + 61T^{2} \) |
| 67 | \( 1 - 8.48T + 67T^{2} \) |
| 71 | \( 1 - 7.90T + 71T^{2} \) |
| 73 | \( 1 - 1.56T + 73T^{2} \) |
| 79 | \( 1 + 8.41T + 79T^{2} \) |
| 83 | \( 1 - 1.55T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 - 0.601T + 97T^{2} \) |
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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
−7.61340544874231501013834892916, −6.89011982997286634456484974176, −6.18069756760889816168222199220, −5.31441971310238912189108754681, −4.59991179645882993791779615186, −3.65962719763827305815909061878, −3.38967293538523982858120221480, −2.38040402410998897991742066653, −1.06835922307571394465595824809, 0,
1.06835922307571394465595824809, 2.38040402410998897991742066653, 3.38967293538523982858120221480, 3.65962719763827305815909061878, 4.59991179645882993791779615186, 5.31441971310238912189108754681, 6.18069756760889816168222199220, 6.89011982997286634456484974176, 7.61340544874231501013834892916
