| L(s) = 1 | − 2·5-s + 2·7-s − 11-s + 4·13-s − 17-s − 2·19-s + 2·23-s − 25-s − 2·29-s + 4·31-s − 4·35-s + 6·37-s − 6·41-s − 2·43-s − 3·49-s + 12·53-s + 2·55-s − 14·59-s + 6·61-s − 8·65-s + 4·67-s − 2·71-s − 8·73-s − 2·77-s + 2·79-s − 12·83-s + 2·85-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.755·7-s − 0.301·11-s + 1.10·13-s − 0.242·17-s − 0.458·19-s + 0.417·23-s − 1/5·25-s − 0.371·29-s + 0.718·31-s − 0.676·35-s + 0.986·37-s − 0.937·41-s − 0.304·43-s − 3/7·49-s + 1.64·53-s + 0.269·55-s − 1.82·59-s + 0.768·61-s − 0.992·65-s + 0.488·67-s − 0.237·71-s − 0.936·73-s − 0.227·77-s + 0.225·79-s − 1.31·83-s + 0.216·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−15.49266857015284, −15.06764731694399, −14.64438053166256, −13.85208805300577, −13.44498885146629, −12.91502798295264, −12.25755005070369, −11.60133110207048, −11.33549746985852, −10.82355460008619, −10.23345688775035, −9.547453807676308, −8.707465327320860, −8.415417677897953, −7.902626837648223, −7.321315777467204, −6.658565161254641, −5.993386602969611, −5.349891425291741, −4.571904859721833, −4.151279300807238, −3.471544197518349, −2.727439645128595, −1.814272611032817, −1.045387240568022, 0,
1.045387240568022, 1.814272611032817, 2.727439645128595, 3.471544197518349, 4.151279300807238, 4.571904859721833, 5.349891425291741, 5.993386602969611, 6.658565161254641, 7.321315777467204, 7.902626837648223, 8.415417677897953, 8.707465327320860, 9.547453807676308, 10.23345688775035, 10.82355460008619, 11.33549746985852, 11.60133110207048, 12.25755005070369, 12.91502798295264, 13.44498885146629, 13.85208805300577, 14.64438053166256, 15.06764731694399, 15.49266857015284
