| L(s) = 1 | + 2-s + 4-s + 2.27·5-s + 8-s + 2.27·10-s + 11-s − 4.63·13-s + 16-s − 0.554·17-s + 4.07·19-s + 2.27·20-s + 22-s + 6.93·23-s + 0.171·25-s − 4.63·26-s + 2.82·29-s + 7.68·31-s + 32-s − 0.554·34-s + 10.8·37-s + 4.07·38-s + 2.27·40-s + 0.554·41-s − 8.59·43-s + 44-s + 6.93·46-s − 10.9·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.01·5-s + 0.353·8-s + 0.719·10-s + 0.301·11-s − 1.28·13-s + 0.250·16-s − 0.134·17-s + 0.935·19-s + 0.508·20-s + 0.213·22-s + 1.44·23-s + 0.0343·25-s − 0.908·26-s + 0.525·29-s + 1.38·31-s + 0.176·32-s − 0.0950·34-s + 1.78·37-s + 0.661·38-s + 0.359·40-s + 0.0865·41-s − 1.31·43-s + 0.150·44-s + 1.02·46-s − 1.59·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.445891901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.445891901\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 2.27T + 5T^{2} \) |
| 13 | \( 1 + 4.63T + 13T^{2} \) |
| 17 | \( 1 + 0.554T + 17T^{2} \) |
| 19 | \( 1 - 4.07T + 19T^{2} \) |
| 23 | \( 1 - 6.93T + 23T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 - 7.68T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 - 0.554T + 41T^{2} \) |
| 43 | \( 1 + 8.59T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 - 0.440T + 53T^{2} \) |
| 59 | \( 1 + 5.17T + 59T^{2} \) |
| 61 | \( 1 - 2.19T + 61T^{2} \) |
| 67 | \( 1 + 11.1T + 67T^{2} \) |
| 71 | \( 1 + 3.60T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 - 15.7T + 79T^{2} \) |
| 83 | \( 1 - 6.51T + 83T^{2} \) |
| 89 | \( 1 - 4.23T + 89T^{2} \) |
| 97 | \( 1 - 3.84T + 97T^{2} \) |
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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
−7.59707159655992327236814887599, −6.67043884478156802575683240672, −6.43013654149151164692283057770, −5.48552228788439949160136406502, −4.94508857249820285055012393814, −4.45145323258622128395809210767, −3.23325929674411569881889109414, −2.73387894166343282327866351614, −1.90018564425422211803507168312, −0.925643581893000526428842730780,
0.925643581893000526428842730780, 1.90018564425422211803507168312, 2.73387894166343282327866351614, 3.23325929674411569881889109414, 4.45145323258622128395809210767, 4.94508857249820285055012393814, 5.48552228788439949160136406502, 6.43013654149151164692283057770, 6.67043884478156802575683240672, 7.59707159655992327236814887599
