| L(s) = 1 | − 3.27·3-s + 3.53·5-s + 2.34·7-s + 7.75·9-s + 11-s − 5.44·13-s − 11.5·15-s + 3.37·17-s + 19-s − 7.68·21-s + 5.78·23-s + 7.49·25-s − 15.5·27-s + 9.19·29-s + 9.59·31-s − 3.27·33-s + 8.28·35-s − 6.58·37-s + 17.8·39-s + 2.37·41-s − 1.18·43-s + 27.4·45-s − 1.69·47-s − 1.50·49-s − 11.0·51-s + 2.51·53-s + 3.53·55-s + ⋯ |
| L(s) = 1 | − 1.89·3-s + 1.58·5-s + 0.886·7-s + 2.58·9-s + 0.301·11-s − 1.50·13-s − 2.99·15-s + 0.818·17-s + 0.229·19-s − 1.67·21-s + 1.20·23-s + 1.49·25-s − 2.99·27-s + 1.70·29-s + 1.72·31-s − 0.570·33-s + 1.40·35-s − 1.08·37-s + 2.85·39-s + 0.370·41-s − 0.179·43-s + 4.08·45-s − 0.246·47-s − 0.214·49-s − 1.55·51-s + 0.344·53-s + 0.476·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.635116363\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.635116363\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 3.27T + 3T^{2} \) |
| 5 | \( 1 - 3.53T + 5T^{2} \) |
| 7 | \( 1 - 2.34T + 7T^{2} \) |
| 13 | \( 1 + 5.44T + 13T^{2} \) |
| 17 | \( 1 - 3.37T + 17T^{2} \) |
| 23 | \( 1 - 5.78T + 23T^{2} \) |
| 29 | \( 1 - 9.19T + 29T^{2} \) |
| 31 | \( 1 - 9.59T + 31T^{2} \) |
| 37 | \( 1 + 6.58T + 37T^{2} \) |
| 41 | \( 1 - 2.37T + 41T^{2} \) |
| 43 | \( 1 + 1.18T + 43T^{2} \) |
| 47 | \( 1 + 1.69T + 47T^{2} \) |
| 53 | \( 1 - 2.51T + 53T^{2} \) |
| 59 | \( 1 - 4.83T + 59T^{2} \) |
| 61 | \( 1 + 7.62T + 61T^{2} \) |
| 67 | \( 1 + 2.22T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 - 5.18T + 79T^{2} \) |
| 83 | \( 1 - 4.24T + 83T^{2} \) |
| 89 | \( 1 - 6.09T + 89T^{2} \) |
| 97 | \( 1 + 6.16T + 97T^{2} \) |
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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
−8.760554871594174469390033441839, −7.56935000542414267931526384492, −6.87750112636822522008438347241, −6.24545587856153396480693651784, −5.56202909156109079122002430128, −4.82559598816350919189715805072, −4.70966561278286143839681752517, −2.81537084908872564234663877899, −1.65086538954313524090494179939, −0.920825191983100859628425416162,
0.920825191983100859628425416162, 1.65086538954313524090494179939, 2.81537084908872564234663877899, 4.70966561278286143839681752517, 4.82559598816350919189715805072, 5.56202909156109079122002430128, 6.24545587856153396480693651784, 6.87750112636822522008438347241, 7.56935000542414267931526384492, 8.760554871594174469390033441839
