| L(s) = 1 | + 3-s + 3.50·5-s + 0.271·7-s + 9-s − 3.15·11-s + 5.54·13-s + 3.50·15-s + 7.90·17-s − 0.447·19-s + 0.271·21-s + 23-s + 7.27·25-s + 27-s − 29-s + 6.65·31-s − 3.15·33-s + 0.952·35-s + 5.73·37-s + 5.54·39-s + 7.82·41-s − 3.05·43-s + 3.50·45-s − 6.56·47-s − 6.92·49-s + 7.90·51-s − 3.63·53-s − 11.0·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.56·5-s + 0.102·7-s + 0.333·9-s − 0.949·11-s + 1.53·13-s + 0.904·15-s + 1.91·17-s − 0.102·19-s + 0.0593·21-s + 0.208·23-s + 1.45·25-s + 0.192·27-s − 0.185·29-s + 1.19·31-s − 0.548·33-s + 0.160·35-s + 0.942·37-s + 0.887·39-s + 1.22·41-s − 0.466·43-s + 0.522·45-s − 0.957·47-s − 0.989·49-s + 1.10·51-s − 0.498·53-s − 1.48·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.268265671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.268265671\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 5 | \( 1 - 3.50T + 5T^{2} \) |
| 7 | \( 1 - 0.271T + 7T^{2} \) |
| 11 | \( 1 + 3.15T + 11T^{2} \) |
| 13 | \( 1 - 5.54T + 13T^{2} \) |
| 17 | \( 1 - 7.90T + 17T^{2} \) |
| 19 | \( 1 + 0.447T + 19T^{2} \) |
| 31 | \( 1 - 6.65T + 31T^{2} \) |
| 37 | \( 1 - 5.73T + 37T^{2} \) |
| 41 | \( 1 - 7.82T + 41T^{2} \) |
| 43 | \( 1 + 3.05T + 43T^{2} \) |
| 47 | \( 1 + 6.56T + 47T^{2} \) |
| 53 | \( 1 + 3.63T + 53T^{2} \) |
| 59 | \( 1 + 9.18T + 59T^{2} \) |
| 61 | \( 1 - 0.878T + 61T^{2} \) |
| 67 | \( 1 + 0.508T + 67T^{2} \) |
| 71 | \( 1 - 7.95T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 + 8.08T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 97 | \( 1 - 2.98T + 97T^{2} \) |
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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
−8.082378453849517453928294721856, −7.18451102794109036195635160709, −6.18882462930089093796360354905, −5.88776101120163056984282326509, −5.18825557042470097666126863597, −4.31337052187724179551958954368, −3.17455604586711436626090871336, −2.80663532576365832296904613241, −1.69204906060381066099086875400, −1.11773143102213837758867721092,
1.11773143102213837758867721092, 1.69204906060381066099086875400, 2.80663532576365832296904613241, 3.17455604586711436626090871336, 4.31337052187724179551958954368, 5.18825557042470097666126863597, 5.88776101120163056984282326509, 6.18882462930089093796360354905, 7.18451102794109036195635160709, 8.082378453849517453928294721856
