| L(s) = 1 | + 2-s + 4-s − 5-s − 5·7-s + 8-s − 3·9-s − 10-s − 3·11-s + 6·13-s − 5·14-s + 16-s − 6·17-s − 3·18-s − 2·19-s − 20-s − 3·22-s − 4·23-s + 25-s + 6·26-s − 5·28-s − 4·31-s + 32-s − 6·34-s + 5·35-s − 3·36-s + 7·37-s − 2·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.88·7-s + 0.353·8-s − 9-s − 0.316·10-s − 0.904·11-s + 1.66·13-s − 1.33·14-s + 1/4·16-s − 1.45·17-s − 0.707·18-s − 0.458·19-s − 0.223·20-s − 0.639·22-s − 0.834·23-s + 1/5·25-s + 1.17·26-s − 0.944·28-s − 0.718·31-s + 0.176·32-s − 1.02·34-s + 0.845·35-s − 1/2·36-s + 1.15·37-s − 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 670 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−10.32845345147070906885501690142, −9.092570694492844195564031890242, −8.432739945518015492258938128722, −7.20828077298441248398619979214, −6.16133957498984848043246819237, −5.88259109198063971715039119953, −4.27652684466614745018881122119, −3.41015423989135969439132788694, −2.54970010423598441604136486775, 0,
2.54970010423598441604136486775, 3.41015423989135969439132788694, 4.27652684466614745018881122119, 5.88259109198063971715039119953, 6.16133957498984848043246819237, 7.20828077298441248398619979214, 8.432739945518015492258938128722, 9.092570694492844195564031890242, 10.32845345147070906885501690142
