L(s) = 1 | + 0.343·2-s − 3-s − 1.88·4-s + 1.20·5-s − 0.343·6-s − 3.15·7-s − 1.33·8-s + 9-s + 0.412·10-s + 1.43·11-s + 1.88·12-s + 13-s − 1.08·14-s − 1.20·15-s + 3.30·16-s − 5.66·17-s + 0.343·18-s + 7.52·19-s − 2.26·20-s + 3.15·21-s + 0.491·22-s − 3.95·23-s + 1.33·24-s − 3.55·25-s + 0.343·26-s − 27-s + 5.93·28-s + ⋯ |
L(s) = 1 | + 0.242·2-s − 0.577·3-s − 0.941·4-s + 0.537·5-s − 0.140·6-s − 1.19·7-s − 0.470·8-s + 0.333·9-s + 0.130·10-s + 0.432·11-s + 0.543·12-s + 0.277·13-s − 0.288·14-s − 0.310·15-s + 0.826·16-s − 1.37·17-s + 0.0808·18-s + 1.72·19-s − 0.505·20-s + 0.687·21-s + 0.104·22-s − 0.823·23-s + 0.271·24-s − 0.711·25-s + 0.0672·26-s − 0.192·27-s + 1.12·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 - 0.343T + 2T^{2} \) |
| 5 | \( 1 - 1.20T + 5T^{2} \) |
| 7 | \( 1 + 3.15T + 7T^{2} \) |
| 11 | \( 1 - 1.43T + 11T^{2} \) |
| 17 | \( 1 + 5.66T + 17T^{2} \) |
| 19 | \( 1 - 7.52T + 19T^{2} \) |
| 23 | \( 1 + 3.95T + 23T^{2} \) |
| 29 | \( 1 - 7.68T + 29T^{2} \) |
| 31 | \( 1 + 1.41T + 31T^{2} \) |
| 37 | \( 1 - 1.97T + 37T^{2} \) |
| 41 | \( 1 - 6.45T + 41T^{2} \) |
| 43 | \( 1 - 7.97T + 43T^{2} \) |
| 47 | \( 1 + 7.42T + 47T^{2} \) |
| 53 | \( 1 + 3.40T + 53T^{2} \) |
| 59 | \( 1 - 1.02T + 59T^{2} \) |
| 61 | \( 1 + 4.32T + 61T^{2} \) |
| 67 | \( 1 + 2.74T + 67T^{2} \) |
| 71 | \( 1 + 1.12T + 71T^{2} \) |
| 73 | \( 1 - 5.18T + 73T^{2} \) |
| 79 | \( 1 + 7.66T + 79T^{2} \) |
| 83 | \( 1 + 3.73T + 83T^{2} \) |
| 89 | \( 1 - 6.74T + 89T^{2} \) |
| 97 | \( 1 - 5.45T + 97T^{2} \) |
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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
−8.131641042666364920403981639969, −7.20270217344733139436050328758, −6.21370928219289237568643575377, −6.05677932025350764014755741109, −5.09217204677310328949061458664, −4.30415236961919507282099324919, −3.57404388592864158848216013004, −2.62785193524250496689505303300, −1.17434728441107897967914662211, 0,
1.17434728441107897967914662211, 2.62785193524250496689505303300, 3.57404388592864158848216013004, 4.30415236961919507282099324919, 5.09217204677310328949061458664, 6.05677932025350764014755741109, 6.21370928219289237568643575377, 7.20270217344733139436050328758, 8.131641042666364920403981639969