| L(s) = 1 | + 5-s + 2·11-s + 2·13-s + 4·19-s − 2·23-s + 25-s + 6·29-s + 2·37-s − 12·41-s + 43-s − 2·47-s − 7·49-s − 8·53-s + 2·55-s + 2·59-s − 2·61-s + 2·65-s − 4·67-s − 8·71-s − 14·73-s − 8·79-s − 14·83-s − 6·89-s + 4·95-s + 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.603·11-s + 0.554·13-s + 0.917·19-s − 0.417·23-s + 1/5·25-s + 1.11·29-s + 0.328·37-s − 1.87·41-s + 0.152·43-s − 0.291·47-s − 49-s − 1.09·53-s + 0.269·55-s + 0.260·59-s − 0.256·61-s + 0.248·65-s − 0.488·67-s − 0.949·71-s − 1.63·73-s − 0.900·79-s − 1.53·83-s − 0.635·89-s + 0.410·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−15.51662542110699, −14.63376260513452, −14.34282302593978, −13.77345581697261, −13.33848075018459, −12.77564014607127, −12.12164254799908, −11.58314759077178, −11.28128174109347, −10.31098581524119, −10.11132859962121, −9.453321397825611, −8.852270864140573, −8.382768784236295, −7.742729758806354, −7.032940681352564, −6.476770943984080, −5.998891590022153, −5.321073282155026, −4.680427092357719, −4.035493701992210, −3.223072916629281, −2.768955188949513, −1.610753090703484, −1.282691340091235, 0,
1.282691340091235, 1.610753090703484, 2.768955188949513, 3.223072916629281, 4.035493701992210, 4.680427092357719, 5.321073282155026, 5.998891590022153, 6.476770943984080, 7.032940681352564, 7.742729758806354, 8.382768784236295, 8.852270864140573, 9.453321397825611, 10.11132859962121, 10.31098581524119, 11.28128174109347, 11.58314759077178, 12.12164254799908, 12.77564014607127, 13.33848075018459, 13.77345581697261, 14.34282302593978, 14.63376260513452, 15.51662542110699
