| L(s) = 1 | − 2.42·3-s + 3.63·5-s − 7-s + 2.90·9-s − 3.86·11-s − 8.83·15-s + 1.63·17-s − 2.01·19-s + 2.42·21-s − 2.41·23-s + 8.23·25-s + 0.233·27-s + 4.02·29-s + 9.12·31-s + 9.39·33-s − 3.63·35-s − 12.0·37-s + 9.56·41-s − 9.86·43-s + 10.5·45-s − 8.94·47-s + 49-s − 3.97·51-s − 6.91·53-s − 14.0·55-s + 4.88·57-s + 0.115·59-s + ⋯ |
| L(s) = 1 | − 1.40·3-s + 1.62·5-s − 0.377·7-s + 0.967·9-s − 1.16·11-s − 2.28·15-s + 0.396·17-s − 0.461·19-s + 0.530·21-s − 0.503·23-s + 1.64·25-s + 0.0449·27-s + 0.748·29-s + 1.63·31-s + 1.63·33-s − 0.614·35-s − 1.97·37-s + 1.49·41-s − 1.50·43-s + 1.57·45-s − 1.30·47-s + 0.142·49-s − 0.556·51-s − 0.950·53-s − 1.89·55-s + 0.647·57-s + 0.0150·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4732 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4732 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 2.42T + 3T^{2} \) |
| 5 | \( 1 - 3.63T + 5T^{2} \) |
| 11 | \( 1 + 3.86T + 11T^{2} \) |
| 17 | \( 1 - 1.63T + 17T^{2} \) |
| 19 | \( 1 + 2.01T + 19T^{2} \) |
| 23 | \( 1 + 2.41T + 23T^{2} \) |
| 29 | \( 1 - 4.02T + 29T^{2} \) |
| 31 | \( 1 - 9.12T + 31T^{2} \) |
| 37 | \( 1 + 12.0T + 37T^{2} \) |
| 41 | \( 1 - 9.56T + 41T^{2} \) |
| 43 | \( 1 + 9.86T + 43T^{2} \) |
| 47 | \( 1 + 8.94T + 47T^{2} \) |
| 53 | \( 1 + 6.91T + 53T^{2} \) |
| 59 | \( 1 - 0.115T + 59T^{2} \) |
| 61 | \( 1 + 4.94T + 61T^{2} \) |
| 67 | \( 1 - 9.11T + 67T^{2} \) |
| 71 | \( 1 - 0.292T + 71T^{2} \) |
| 73 | \( 1 + 1.89T + 73T^{2} \) |
| 79 | \( 1 - 5.12T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 + 2.46T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−7.920625300912511502909550997279, −6.73882712828888178892444510646, −6.40135606035435276676046584045, −5.74866528573909012576265550915, −5.16539232886272105698109290023, −4.64211836707282827669446897843, −3.16189032925171636683374775300, −2.29961597734273650810113677846, −1.27725833693979325442039651790, 0,
1.27725833693979325442039651790, 2.29961597734273650810113677846, 3.16189032925171636683374775300, 4.64211836707282827669446897843, 5.16539232886272105698109290023, 5.74866528573909012576265550915, 6.40135606035435276676046584045, 6.73882712828888178892444510646, 7.920625300912511502909550997279
