L(s) = 1 | − 2-s + 4-s + 2·7-s − 8-s − 4·11-s + 13-s − 2·14-s + 16-s + 4·17-s − 2·19-s + 4·22-s + 2·23-s − 26-s + 2·28-s − 8·29-s + 4·31-s − 32-s − 4·34-s − 6·37-s + 2·38-s − 10·41-s − 4·43-s − 4·44-s − 2·46-s − 3·49-s + 52-s + 6·53-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s − 1.20·11-s + 0.277·13-s − 0.534·14-s + 1/4·16-s + 0.970·17-s − 0.458·19-s + 0.852·22-s + 0.417·23-s − 0.196·26-s + 0.377·28-s − 1.48·29-s + 0.718·31-s − 0.176·32-s − 0.685·34-s − 0.986·37-s + 0.324·38-s − 1.56·41-s − 0.609·43-s − 0.603·44-s − 0.294·46-s − 3/7·49-s + 0.138·52-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−7.908132220951708075530149837207, −7.22411220646260145593997176085, −6.50035574002823002786913644509, −5.36386658029537350392734221766, −5.21970400813546853897790995357, −3.96122346803787745690914824334, −3.09094631690423310682187162551, −2.15589679964419797750219216774, −1.31446428753136108036304683912, 0,
1.31446428753136108036304683912, 2.15589679964419797750219216774, 3.09094631690423310682187162551, 3.96122346803787745690914824334, 5.21970400813546853897790995357, 5.36386658029537350392734221766, 6.50035574002823002786913644509, 7.22411220646260145593997176085, 7.908132220951708075530149837207