| L(s) = 1 | − 0.381·2-s − 1.85·4-s + 5-s − 7-s + 1.47·8-s − 0.381·10-s − 0.236·13-s + 0.381·14-s + 3.14·16-s − 6.47·17-s − 3.47·19-s − 1.85·20-s + 8.47·23-s − 4·25-s + 0.0901·26-s + 1.85·28-s − 1.76·29-s − 0.472·31-s − 4.14·32-s + 2.47·34-s − 35-s − 5.47·37-s + 1.32·38-s + 1.47·40-s − 4.47·41-s + 4·43-s − 3.23·46-s + ⋯ |
| L(s) = 1 | − 0.270·2-s − 0.927·4-s + 0.447·5-s − 0.377·7-s + 0.520·8-s − 0.120·10-s − 0.0654·13-s + 0.102·14-s + 0.786·16-s − 1.56·17-s − 0.796·19-s − 0.414·20-s + 1.76·23-s − 0.800·25-s + 0.0176·26-s + 0.350·28-s − 0.327·29-s − 0.0847·31-s − 0.732·32-s + 0.423·34-s − 0.169·35-s − 0.899·37-s + 0.215·38-s + 0.232·40-s − 0.698·41-s + 0.609·43-s − 0.477·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9456409690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9456409690\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.381T + 2T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 13 | \( 1 + 0.236T + 13T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 + 3.47T + 19T^{2} \) |
| 23 | \( 1 - 8.47T + 23T^{2} \) |
| 29 | \( 1 + 1.76T + 29T^{2} \) |
| 31 | \( 1 + 0.472T + 31T^{2} \) |
| 37 | \( 1 + 5.47T + 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 0.236T + 47T^{2} \) |
| 53 | \( 1 - 5.52T + 53T^{2} \) |
| 59 | \( 1 - 15.1T + 59T^{2} \) |
| 61 | \( 1 + 8.47T + 61T^{2} \) |
| 67 | \( 1 - 9.18T + 67T^{2} \) |
| 71 | \( 1 + 4.47T + 71T^{2} \) |
| 73 | \( 1 - 9.76T + 73T^{2} \) |
| 79 | \( 1 + 4.47T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−8.072537442315104861818779814518, −7.03812140418263025581557444131, −6.66671550612483106125045788763, −5.66539276884666793242116601304, −5.09058976147054486344967319685, −4.30746572293853628224035602271, −3.67371225275942759703907226113, −2.61038805662279907323391203982, −1.73013334471359669612718285780, −0.50696692675460602036342334643,
0.50696692675460602036342334643, 1.73013334471359669612718285780, 2.61038805662279907323391203982, 3.67371225275942759703907226113, 4.30746572293853628224035602271, 5.09058976147054486344967319685, 5.66539276884666793242116601304, 6.66671550612483106125045788763, 7.03812140418263025581557444131, 8.072537442315104861818779814518
