| L(s) = 1 | + 2.41·2-s + 3.82·4-s + 2.82·5-s + 4.41·8-s + 6.82·10-s + 4.82·11-s + 4.82·13-s + 2.99·16-s + 1.17·17-s + 19-s + 10.8·20-s + 11.6·22-s + 4.82·23-s + 3.00·25-s + 11.6·26-s − 8.48·29-s − 1.58·32-s + 2.82·34-s − 6·37-s + 2.41·38-s + 12.4·40-s + 5.65·41-s + 18.4·44-s + 11.6·46-s − 8.82·47-s + 7.24·50-s + 18.4·52-s + ⋯ |
| L(s) = 1 | + 1.70·2-s + 1.91·4-s + 1.26·5-s + 1.56·8-s + 2.15·10-s + 1.45·11-s + 1.33·13-s + 0.749·16-s + 0.284·17-s + 0.229·19-s + 2.42·20-s + 2.48·22-s + 1.00·23-s + 0.600·25-s + 2.28·26-s − 1.57·29-s − 0.280·32-s + 0.485·34-s − 0.986·37-s + 0.391·38-s + 1.97·40-s + 0.883·41-s + 2.78·44-s + 1.71·46-s − 1.28·47-s + 1.02·50-s + 2.56·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8379 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.287139503\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.287139503\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.41T + 2T^{2} \) |
| 5 | \( 1 - 2.82T + 5T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 - 4.82T + 13T^{2} \) |
| 17 | \( 1 - 1.17T + 17T^{2} \) |
| 23 | \( 1 - 4.82T + 23T^{2} \) |
| 29 | \( 1 + 8.48T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 5.65T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 8.82T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 5.65T + 59T^{2} \) |
| 61 | \( 1 - 4T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 4.34T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 - 2.48T + 79T^{2} \) |
| 83 | \( 1 - 4.82T + 83T^{2} \) |
| 89 | \( 1 + 17.6T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−7.40485734412077937150370935600, −6.63540691769122829687294287381, −6.25991864105311545734733336830, −5.67750786425896670368784004197, −5.12468948513043959496554063908, −4.22377232870271963348566590360, −3.57453954600856998698479638833, −2.98968921574449153862325786165, −1.82179982687323108952470841901, −1.38551194748689730126551863629,
1.38551194748689730126551863629, 1.82179982687323108952470841901, 2.98968921574449153862325786165, 3.57453954600856998698479638833, 4.22377232870271963348566590360, 5.12468948513043959496554063908, 5.67750786425896670368784004197, 6.25991864105311545734733336830, 6.63540691769122829687294287381, 7.40485734412077937150370935600
