| L(s) = 1 | + 3.44·3-s − 1.05·5-s + 7-s + 8.83·9-s + 5.76·11-s + 13-s − 3.61·15-s + 2.81·17-s − 1.60·19-s + 3.44·21-s + 3.19·23-s − 3.89·25-s + 20.0·27-s + 0.0108·29-s + 1.42·31-s + 19.8·33-s − 1.05·35-s + 6.32·37-s + 3.44·39-s − 5.75·41-s − 7.02·43-s − 9.28·45-s − 7.30·47-s + 49-s + 9.68·51-s − 13.8·53-s − 6.06·55-s + ⋯ |
| L(s) = 1 | + 1.98·3-s − 0.470·5-s + 0.377·7-s + 2.94·9-s + 1.73·11-s + 0.277·13-s − 0.933·15-s + 0.682·17-s − 0.368·19-s + 0.750·21-s + 0.665·23-s − 0.778·25-s + 3.86·27-s + 0.00200·29-s + 0.256·31-s + 3.45·33-s − 0.177·35-s + 1.03·37-s + 0.550·39-s − 0.899·41-s − 1.07·43-s − 1.38·45-s − 1.06·47-s + 0.142·49-s + 1.35·51-s − 1.89·53-s − 0.817·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5824 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5824 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.123310785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.123310785\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 3.44T + 3T^{2} \) |
| 5 | \( 1 + 1.05T + 5T^{2} \) |
| 11 | \( 1 - 5.76T + 11T^{2} \) |
| 17 | \( 1 - 2.81T + 17T^{2} \) |
| 19 | \( 1 + 1.60T + 19T^{2} \) |
| 23 | \( 1 - 3.19T + 23T^{2} \) |
| 29 | \( 1 - 0.0108T + 29T^{2} \) |
| 31 | \( 1 - 1.42T + 31T^{2} \) |
| 37 | \( 1 - 6.32T + 37T^{2} \) |
| 41 | \( 1 + 5.75T + 41T^{2} \) |
| 43 | \( 1 + 7.02T + 43T^{2} \) |
| 47 | \( 1 + 7.30T + 47T^{2} \) |
| 53 | \( 1 + 13.8T + 53T^{2} \) |
| 59 | \( 1 + 8.08T + 59T^{2} \) |
| 61 | \( 1 - 5.75T + 61T^{2} \) |
| 67 | \( 1 + 9.71T + 67T^{2} \) |
| 71 | \( 1 + 3.61T + 71T^{2} \) |
| 73 | \( 1 + 2.95T + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 + 3.88T + 83T^{2} \) |
| 89 | \( 1 - 12.0T + 89T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−8.119755871876924337268250604675, −7.69884491700693629676825765764, −6.85725816102435399903944768742, −6.30039902817910374706060918264, −4.82306117258470983983769396397, −4.19280974010238579783681123168, −3.54146664457019283155142467354, −3.01670043200214155895134870513, −1.81271526754453131509414928210, −1.28012134695305910475548471707,
1.28012134695305910475548471707, 1.81271526754453131509414928210, 3.01670043200214155895134870513, 3.54146664457019283155142467354, 4.19280974010238579783681123168, 4.82306117258470983983769396397, 6.30039902817910374706060918264, 6.85725816102435399903944768742, 7.69884491700693629676825765764, 8.119755871876924337268250604675
