| L(s) = 1 | − 2-s + 4-s + 1.09·5-s − 8-s − 1.09·10-s − 11-s − 6.80·13-s + 16-s + 1.09·17-s + 3.89·19-s + 1.09·20-s + 22-s + 6.99·23-s − 3.80·25-s + 6.80·26-s − 4.80·29-s + 1.27·31-s − 32-s − 1.09·34-s + 4.18·37-s − 3.89·38-s − 1.09·40-s − 1.09·41-s + 10.6·43-s − 44-s − 6.99·46-s − 9.71·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.488·5-s − 0.353·8-s − 0.345·10-s − 0.301·11-s − 1.88·13-s + 0.250·16-s + 0.264·17-s + 0.894·19-s + 0.244·20-s + 0.213·22-s + 1.45·23-s − 0.761·25-s + 1.33·26-s − 0.892·29-s + 0.229·31-s − 0.176·32-s − 0.187·34-s + 0.687·37-s − 0.632·38-s − 0.172·40-s − 0.170·41-s + 1.62·43-s − 0.150·44-s − 1.03·46-s − 1.41·47-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
| \(L(1)\) |
\(\approx\) |
\(1.259693767\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.259693767\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.09T + 5T^{2} \) |
| 13 | \( 1 + 6.80T + 13T^{2} \) |
| 17 | \( 1 - 1.09T + 17T^{2} \) |
| 19 | \( 1 - 3.89T + 19T^{2} \) |
| 23 | \( 1 - 6.99T + 23T^{2} \) |
| 29 | \( 1 + 4.80T + 29T^{2} \) |
| 31 | \( 1 - 1.27T + 31T^{2} \) |
| 37 | \( 1 - 4.18T + 37T^{2} \) |
| 41 | \( 1 + 1.09T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 + 9.71T + 47T^{2} \) |
| 53 | \( 1 + 3.81T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 - 8.99T + 67T^{2} \) |
| 71 | \( 1 - 9.17T + 71T^{2} \) |
| 73 | \( 1 - 5.09T + 73T^{2} \) |
| 79 | \( 1 + 4.99T + 79T^{2} \) |
| 83 | \( 1 + 7.09T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 7.61T + 97T^{2} \) |
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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
−7.64391728936472884661996537496, −7.18102637658812453981990510914, −6.50650257661527873657635893897, −5.48277430109814115105759310119, −5.21191890374044697151146024502, −4.23820828054082575753576885939, −3.09009822359299063808755336115, −2.54058653874809445673165408209, −1.69094439971531971344752745872, −0.58769851585255669049142137115,
0.58769851585255669049142137115, 1.69094439971531971344752745872, 2.54058653874809445673165408209, 3.09009822359299063808755336115, 4.23820828054082575753576885939, 5.21191890374044697151146024502, 5.48277430109814115105759310119, 6.50650257661527873657635893897, 7.18102637658812453981990510914, 7.64391728936472884661996537496
