| L(s) = 1 | − 1.49·3-s + 1.02·5-s − 3.87·7-s − 0.770·9-s + 4.61·11-s − 4.62·13-s − 1.53·15-s + 2.47·17-s + 7.75·19-s + 5.78·21-s − 2.50·23-s − 3.94·25-s + 5.63·27-s − 4.13·29-s − 1.95·31-s − 6.89·33-s − 3.98·35-s − 1.50·37-s + 6.90·39-s + 6.64·41-s − 3.59·43-s − 0.791·45-s − 47-s + 8.03·49-s − 3.69·51-s − 10.4·53-s + 4.74·55-s + ⋯ |
| L(s) = 1 | − 0.861·3-s + 0.459·5-s − 1.46·7-s − 0.256·9-s + 1.39·11-s − 1.28·13-s − 0.396·15-s + 0.599·17-s + 1.77·19-s + 1.26·21-s − 0.522·23-s − 0.788·25-s + 1.08·27-s − 0.767·29-s − 0.350·31-s − 1.19·33-s − 0.673·35-s − 0.246·37-s + 1.10·39-s + 1.03·41-s − 0.547·43-s − 0.118·45-s − 0.145·47-s + 1.14·49-s − 0.516·51-s − 1.43·53-s + 0.639·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9129213440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9129213440\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 47 | \( 1 + T \) |
| good | 3 | \( 1 + 1.49T + 3T^{2} \) |
| 5 | \( 1 - 1.02T + 5T^{2} \) |
| 7 | \( 1 + 3.87T + 7T^{2} \) |
| 11 | \( 1 - 4.61T + 11T^{2} \) |
| 13 | \( 1 + 4.62T + 13T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 19 | \( 1 - 7.75T + 19T^{2} \) |
| 23 | \( 1 + 2.50T + 23T^{2} \) |
| 29 | \( 1 + 4.13T + 29T^{2} \) |
| 31 | \( 1 + 1.95T + 31T^{2} \) |
| 37 | \( 1 + 1.50T + 37T^{2} \) |
| 41 | \( 1 - 6.64T + 41T^{2} \) |
| 43 | \( 1 + 3.59T + 43T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 - 5.25T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 + 15.4T + 67T^{2} \) |
| 71 | \( 1 - 6.38T + 71T^{2} \) |
| 73 | \( 1 - 2.34T + 73T^{2} \) |
| 79 | \( 1 - 15.0T + 79T^{2} \) |
| 83 | \( 1 + 2.24T + 83T^{2} \) |
| 89 | \( 1 + 7.54T + 89T^{2} \) |
| 97 | \( 1 + 3.03T + 97T^{2} \) |
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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
−7.87627892661642679768629828790, −7.19353831525142800382656251253, −6.54040418226597796196551052055, −5.89212184589901133989332405999, −5.50466406383406387031076811922, −4.54768451053893810330762454282, −3.50316457758330528865531948517, −2.96958167924358077808271679571, −1.72114556704411200891034265285, −0.51928399731838092882394064640,
0.51928399731838092882394064640, 1.72114556704411200891034265285, 2.96958167924358077808271679571, 3.50316457758330528865531948517, 4.54768451053893810330762454282, 5.50466406383406387031076811922, 5.89212184589901133989332405999, 6.54040418226597796196551052055, 7.19353831525142800382656251253, 7.87627892661642679768629828790
