L(s) = 1 | + 3-s − 5-s + 4·7-s + 9-s − 2·11-s − 6·13-s − 15-s − 2·17-s + 4·21-s − 6·23-s + 25-s + 27-s − 8·29-s − 8·31-s − 2·33-s − 4·35-s − 10·37-s − 6·39-s + 4·41-s − 4·43-s − 45-s − 6·47-s + 9·49-s − 2·51-s − 12·53-s + 2·55-s − 8·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.603·11-s − 1.66·13-s − 0.258·15-s − 0.485·17-s + 0.872·21-s − 1.25·23-s + 1/5·25-s + 0.192·27-s − 1.48·29-s − 1.43·31-s − 0.348·33-s − 0.676·35-s − 1.64·37-s − 0.960·39-s + 0.624·41-s − 0.609·43-s − 0.149·45-s − 0.875·47-s + 9/7·49-s − 0.280·51-s − 1.64·53-s + 0.269·55-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7948190705\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7948190705\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−14.17633893074109, −13.42268884063639, −12.89945636665329, −12.43320032436218, −11.93663631588431, −11.43419898701353, −10.95176027461753, −10.51306874296345, −9.857993817595649, −9.396248373391720, −8.816548112960635, −8.246985784344063, −7.809452790645293, −7.450618744900147, −7.103467080098868, −6.208604803862543, −5.418489492464161, −4.973228471701792, −4.634277498972228, −3.899789355110170, −3.369522045027907, −2.530698891478628, −1.828544029035952, −1.749136156060141, −0.2505947682201484,
0.2505947682201484, 1.749136156060141, 1.828544029035952, 2.530698891478628, 3.369522045027907, 3.899789355110170, 4.634277498972228, 4.973228471701792, 5.418489492464161, 6.208604803862543, 7.103467080098868, 7.450618744900147, 7.809452790645293, 8.246985784344063, 8.816548112960635, 9.396248373391720, 9.857993817595649, 10.51306874296345, 10.95176027461753, 11.43419898701353, 11.93663631588431, 12.43320032436218, 12.89945636665329, 13.42268884063639, 14.17633893074109