| L(s) = 1 | + 2.60·3-s + 1.12·5-s − 1.15·7-s + 3.77·9-s + 4.62·11-s + 6.29·13-s + 2.93·15-s − 4.65·17-s + 4.55·19-s − 2.99·21-s + 8.09·23-s − 3.72·25-s + 2.00·27-s − 4.35·29-s − 3.23·31-s + 12.0·33-s − 1.30·35-s − 1.34·37-s + 16.3·39-s − 0.554·41-s + 2.45·43-s + 4.25·45-s − 4.98·47-s − 5.67·49-s − 12.1·51-s + 11.5·53-s + 5.22·55-s + ⋯ |
| L(s) = 1 | + 1.50·3-s + 0.505·5-s − 0.435·7-s + 1.25·9-s + 1.39·11-s + 1.74·13-s + 0.758·15-s − 1.12·17-s + 1.04·19-s − 0.653·21-s + 1.68·23-s − 0.744·25-s + 0.386·27-s − 0.808·29-s − 0.580·31-s + 2.09·33-s − 0.219·35-s − 0.221·37-s + 2.62·39-s − 0.0866·41-s + 0.374·43-s + 0.634·45-s − 0.726·47-s − 0.810·49-s − 1.69·51-s + 1.58·53-s + 0.704·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.201482712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.201482712\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 2.60T + 3T^{2} \) |
| 5 | \( 1 - 1.12T + 5T^{2} \) |
| 7 | \( 1 + 1.15T + 7T^{2} \) |
| 11 | \( 1 - 4.62T + 11T^{2} \) |
| 13 | \( 1 - 6.29T + 13T^{2} \) |
| 17 | \( 1 + 4.65T + 17T^{2} \) |
| 19 | \( 1 - 4.55T + 19T^{2} \) |
| 23 | \( 1 - 8.09T + 23T^{2} \) |
| 29 | \( 1 + 4.35T + 29T^{2} \) |
| 31 | \( 1 + 3.23T + 31T^{2} \) |
| 37 | \( 1 + 1.34T + 37T^{2} \) |
| 41 | \( 1 + 0.554T + 41T^{2} \) |
| 43 | \( 1 - 2.45T + 43T^{2} \) |
| 47 | \( 1 + 4.98T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 8.43T + 59T^{2} \) |
| 61 | \( 1 + 0.328T + 61T^{2} \) |
| 67 | \( 1 - 2.55T + 67T^{2} \) |
| 71 | \( 1 + 9.67T + 71T^{2} \) |
| 73 | \( 1 + 1.53T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 - 4.49T + 83T^{2} \) |
| 89 | \( 1 - 13.9T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−8.825809018949878330759385695432, −7.80238701726655520641664275602, −7.01041581697801607636484591318, −6.38968419961421702093472857658, −5.59444500929931294822980607526, −4.34260850220429536327474263858, −3.58097455551398436958152916333, −3.14761737542207760186464110109, −1.97080785240180339006860809662, −1.24050634206396379431514488276,
1.24050634206396379431514488276, 1.97080785240180339006860809662, 3.14761737542207760186464110109, 3.58097455551398436958152916333, 4.34260850220429536327474263858, 5.59444500929931294822980607526, 6.38968419961421702093472857658, 7.01041581697801607636484591318, 7.80238701726655520641664275602, 8.825809018949878330759385695432
