| L(s) = 1 | − 3.42·3-s + 2.35·5-s + 8.72·9-s − 0.142·11-s − 2.89·13-s − 8.07·15-s − 1.39·17-s − 7.23·19-s − 4.70·23-s + 0.557·25-s − 19.5·27-s + 2.50·29-s − 2.29·31-s + 0.489·33-s − 8.95·37-s + 9.89·39-s − 41-s + 10.0·43-s + 20.5·45-s − 0.577·47-s + 4.77·51-s − 2.67·53-s − 0.336·55-s + 24.7·57-s − 5.00·59-s + 10.1·61-s − 6.81·65-s + ⋯ |
| L(s) = 1 | − 1.97·3-s + 1.05·5-s + 2.90·9-s − 0.0430·11-s − 0.801·13-s − 2.08·15-s − 0.338·17-s − 1.65·19-s − 0.980·23-s + 0.111·25-s − 3.77·27-s + 0.464·29-s − 0.412·31-s + 0.0851·33-s − 1.47·37-s + 1.58·39-s − 0.156·41-s + 1.53·43-s + 3.06·45-s − 0.0842·47-s + 0.668·51-s − 0.367·53-s − 0.0454·55-s + 3.28·57-s − 0.651·59-s + 1.30·61-s − 0.845·65-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
| \(L(1)\) |
\(\approx\) |
\(0.6886208915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6886208915\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.42T + 3T^{2} \) |
| 5 | \( 1 - 2.35T + 5T^{2} \) |
| 11 | \( 1 + 0.142T + 11T^{2} \) |
| 13 | \( 1 + 2.89T + 13T^{2} \) |
| 17 | \( 1 + 1.39T + 17T^{2} \) |
| 19 | \( 1 + 7.23T + 19T^{2} \) |
| 23 | \( 1 + 4.70T + 23T^{2} \) |
| 29 | \( 1 - 2.50T + 29T^{2} \) |
| 31 | \( 1 + 2.29T + 31T^{2} \) |
| 37 | \( 1 + 8.95T + 37T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 + 0.577T + 47T^{2} \) |
| 53 | \( 1 + 2.67T + 53T^{2} \) |
| 59 | \( 1 + 5.00T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 - 4.81T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 + 2.06T + 73T^{2} \) |
| 79 | \( 1 - 2.74T + 79T^{2} \) |
| 83 | \( 1 - 8.58T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 + 4.73T + 97T^{2} \) |
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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
−7.49955940169038259391773342148, −6.89226135818552444570843575988, −6.19605243961153561323860038891, −5.92015099547153351628652550831, −5.13234733218789894804467953006, −4.59046687822490918929420048862, −3.85538207472432115806771460970, −2.23933095695166641313976802638, −1.73079340007142669518234532208, −0.44737255001197887266123154107,
0.44737255001197887266123154107, 1.73079340007142669518234532208, 2.23933095695166641313976802638, 3.85538207472432115806771460970, 4.59046687822490918929420048862, 5.13234733218789894804467953006, 5.92015099547153351628652550831, 6.19605243961153561323860038891, 6.89226135818552444570843575988, 7.49955940169038259391773342148
