| L(s) = 1 | + 0.414·2-s − 1.82·4-s + 2.82·5-s − 3·7-s − 1.58·8-s + 1.17·10-s + 4.82·11-s − 4.65·13-s − 1.24·14-s + 3·16-s − 3·19-s − 5.17·20-s + 1.99·22-s + 2.82·23-s + 3.00·25-s − 1.92·26-s + 5.48·28-s − 8.82·29-s − 6.65·31-s + 4.41·32-s − 8.48·35-s + 4.65·37-s − 1.24·38-s − 4.48·40-s + 3.17·41-s + 3·43-s − 8.82·44-s + ⋯ |
| L(s) = 1 | + 0.292·2-s − 0.914·4-s + 1.26·5-s − 1.13·7-s − 0.560·8-s + 0.370·10-s + 1.45·11-s − 1.29·13-s − 0.332·14-s + 0.750·16-s − 0.688·19-s − 1.15·20-s + 0.426·22-s + 0.589·23-s + 0.600·25-s − 0.378·26-s + 1.03·28-s − 1.63·29-s − 1.19·31-s + 0.780·32-s − 1.43·35-s + 0.765·37-s − 0.201·38-s − 0.709·40-s + 0.495·41-s + 0.457·43-s − 1.33·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 0.414T + 2T^{2} \) |
| 5 | \( 1 - 2.82T + 5T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - 4.82T + 11T^{2} \) |
| 13 | \( 1 + 4.65T + 13T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 8.82T + 29T^{2} \) |
| 31 | \( 1 + 6.65T + 31T^{2} \) |
| 37 | \( 1 - 4.65T + 37T^{2} \) |
| 41 | \( 1 - 3.17T + 41T^{2} \) |
| 43 | \( 1 - 3T + 43T^{2} \) |
| 47 | \( 1 + 1.65T + 47T^{2} \) |
| 53 | \( 1 + 9.65T + 53T^{2} \) |
| 59 | \( 1 - 3.17T + 59T^{2} \) |
| 61 | \( 1 + 6.65T + 61T^{2} \) |
| 67 | \( 1 - T + 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 + 3.65T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 + 6.82T + 83T^{2} \) |
| 89 | \( 1 - 17.3T + 89T^{2} \) |
| 97 | \( 1 + 0.656T + 97T^{2} \) |
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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
−9.028303091962508915209678269616, −7.64023381984453899902917846840, −6.74929136559912823282056237142, −6.06230832795072243431084000379, −5.49459510061699868608810406753, −4.50232029794972677023856294930, −3.70982402960184216360753589386, −2.74714204635400215149953393905, −1.58871635989500288686878782469, 0,
1.58871635989500288686878782469, 2.74714204635400215149953393905, 3.70982402960184216360753589386, 4.50232029794972677023856294930, 5.49459510061699868608810406753, 6.06230832795072243431084000379, 6.74929136559912823282056237142, 7.64023381984453899902917846840, 9.028303091962508915209678269616
