| L(s) = 1 | + 2-s − 0.835·3-s + 4-s − 3.09·5-s − 0.835·6-s + 4.05·7-s + 8-s − 2.30·9-s − 3.09·10-s − 2.34·11-s − 0.835·12-s − 6.11·13-s + 4.05·14-s + 2.58·15-s + 16-s − 4.64·17-s − 2.30·18-s + 7.87·19-s − 3.09·20-s − 3.38·21-s − 2.34·22-s − 4.52·23-s − 0.835·24-s + 4.59·25-s − 6.11·26-s + 4.42·27-s + 4.05·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.482·3-s + 0.5·4-s − 1.38·5-s − 0.341·6-s + 1.53·7-s + 0.353·8-s − 0.767·9-s − 0.979·10-s − 0.706·11-s − 0.241·12-s − 1.69·13-s + 1.08·14-s + 0.668·15-s + 0.250·16-s − 1.12·17-s − 0.542·18-s + 1.80·19-s − 0.692·20-s − 0.738·21-s − 0.499·22-s − 0.943·23-s − 0.170·24-s + 0.918·25-s − 1.19·26-s + 0.852·27-s + 0.766·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.460559648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.460559648\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 + 0.835T + 3T^{2} \) |
| 5 | \( 1 + 3.09T + 5T^{2} \) |
| 7 | \( 1 - 4.05T + 7T^{2} \) |
| 11 | \( 1 + 2.34T + 11T^{2} \) |
| 13 | \( 1 + 6.11T + 13T^{2} \) |
| 17 | \( 1 + 4.64T + 17T^{2} \) |
| 19 | \( 1 - 7.87T + 19T^{2} \) |
| 23 | \( 1 + 4.52T + 23T^{2} \) |
| 29 | \( 1 + 3.24T + 29T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 + 7.00T + 41T^{2} \) |
| 43 | \( 1 + 6.98T + 43T^{2} \) |
| 47 | \( 1 - 2.60T + 47T^{2} \) |
| 53 | \( 1 - 3.90T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 - 2.75T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 - 13.5T + 73T^{2} \) |
| 79 | \( 1 - 3.22T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 + 8.84T + 89T^{2} \) |
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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
−7.958075657400519690889119048031, −7.44879923965560428588111869135, −6.77596116682563109151311242863, −5.55880620247401308934792227330, −5.10133429234325475129459030816, −4.63340919821998428163663632802, −3.84977239682426764851654176546, −2.81959268536023527843423066729, −2.08573194410171004338809184274, −0.55878424386190746740165797760,
0.55878424386190746740165797760, 2.08573194410171004338809184274, 2.81959268536023527843423066729, 3.84977239682426764851654176546, 4.63340919821998428163663632802, 5.10133429234325475129459030816, 5.55880620247401308934792227330, 6.77596116682563109151311242863, 7.44879923965560428588111869135, 7.958075657400519690889119048031
