| L(s) = 1 | + 4·13-s + 2·17-s − 8·19-s − 8·23-s − 8·29-s + 4·31-s + 8·37-s + 12·41-s + 8·43-s − 4·47-s − 6·53-s − 8·59-s + 6·61-s + 8·67-s − 4·73-s − 8·79-s − 4·89-s − 12·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.10·13-s + 0.485·17-s − 1.83·19-s − 1.66·23-s − 1.48·29-s + 0.718·31-s + 1.31·37-s + 1.87·41-s + 1.21·43-s − 0.583·47-s − 0.824·53-s − 1.04·59-s + 0.768·61-s + 0.977·67-s − 0.468·73-s − 0.900·79-s − 0.423·89-s − 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.915158020\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.915158020\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−13.06027683435981, −12.71252500043484, −12.45193779950927, −11.62649504990045, −11.25400850957063, −10.92227011014102, −10.34580598678871, −9.917580524976989, −9.283021055243000, −8.989517687884215, −8.230233411318694, −7.982893439211754, −7.565348200292929, −6.756427093857477, −6.250306727498229, −5.883450698507972, −5.583322162730049, −4.487778819382102, −4.261619061263452, −3.808590952751502, −3.106039552003228, −2.366181162505625, −1.934525157162055, −1.186890665193411, −0.4133708181216057,
0.4133708181216057, 1.186890665193411, 1.934525157162055, 2.366181162505625, 3.106039552003228, 3.808590952751502, 4.261619061263452, 4.487778819382102, 5.583322162730049, 5.883450698507972, 6.250306727498229, 6.756427093857477, 7.565348200292929, 7.982893439211754, 8.230233411318694, 8.989517687884215, 9.283021055243000, 9.917580524976989, 10.34580598678871, 10.92227011014102, 11.25400850957063, 11.62649504990045, 12.45193779950927, 12.71252500043484, 13.06027683435981
