| L(s) = 1 | + 2.17·2-s + 2.70·4-s + 0.539·7-s + 1.53·8-s − 3.17·11-s − 4.87·13-s + 1.17·14-s − 2.07·16-s + 1.36·17-s − 19-s − 6.87·22-s − 2.78·23-s − 10.5·26-s + 1.46·28-s − 3.90·29-s − 2.44·31-s − 7.58·32-s + 2.97·34-s + 4.14·37-s − 2.17·38-s − 3.01·41-s − 5.95·43-s − 8.58·44-s − 6.04·46-s − 4.04·47-s − 6.70·49-s − 13.2·52-s + ⋯ |
| L(s) = 1 | + 1.53·2-s + 1.35·4-s + 0.203·7-s + 0.544·8-s − 0.955·11-s − 1.35·13-s + 0.312·14-s − 0.519·16-s + 0.332·17-s − 0.229·19-s − 1.46·22-s − 0.581·23-s − 2.07·26-s + 0.276·28-s − 0.725·29-s − 0.439·31-s − 1.34·32-s + 0.509·34-s + 0.680·37-s − 0.352·38-s − 0.470·41-s − 0.908·43-s − 1.29·44-s − 0.891·46-s − 0.590·47-s − 0.958·49-s − 1.83·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 2.17T + 2T^{2} \) |
| 7 | \( 1 - 0.539T + 7T^{2} \) |
| 11 | \( 1 + 3.17T + 11T^{2} \) |
| 13 | \( 1 + 4.87T + 13T^{2} \) |
| 17 | \( 1 - 1.36T + 17T^{2} \) |
| 23 | \( 1 + 2.78T + 23T^{2} \) |
| 29 | \( 1 + 3.90T + 29T^{2} \) |
| 31 | \( 1 + 2.44T + 31T^{2} \) |
| 37 | \( 1 - 4.14T + 37T^{2} \) |
| 41 | \( 1 + 3.01T + 41T^{2} \) |
| 43 | \( 1 + 5.95T + 43T^{2} \) |
| 47 | \( 1 + 4.04T + 47T^{2} \) |
| 53 | \( 1 - 6.63T + 53T^{2} \) |
| 59 | \( 1 - 12.4T + 59T^{2} \) |
| 61 | \( 1 + 9.31T + 61T^{2} \) |
| 67 | \( 1 - 7.75T + 67T^{2} \) |
| 71 | \( 1 + 2.18T + 71T^{2} \) |
| 73 | \( 1 - 7.60T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 - 2.78T + 83T^{2} \) |
| 89 | \( 1 + 0.829T + 89T^{2} \) |
| 97 | \( 1 - 5.37T + 97T^{2} \) |
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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
−7.76972494556269389559350839937, −7.16110666571387460077964185863, −6.35921151406188504656517756037, −5.47403888093077232248924606700, −5.09188028724073876435190729329, −4.35205604185464992141291367613, −3.49473348715151351377426296007, −2.65819180016240587466074530413, −1.95155425807357077389663745189, 0,
1.95155425807357077389663745189, 2.65819180016240587466074530413, 3.49473348715151351377426296007, 4.35205604185464992141291367613, 5.09188028724073876435190729329, 5.47403888093077232248924606700, 6.35921151406188504656517756037, 7.16110666571387460077964185863, 7.76972494556269389559350839937
