| L(s) = 1 | + 5-s − 6·13-s + 2·17-s − 2·19-s + 6·23-s + 25-s + 6·29-s − 4·31-s − 8·37-s + 8·41-s − 43-s + 6·47-s − 7·49-s + 6·53-s + 4·59-s − 14·61-s − 6·65-s + 4·67-s − 8·71-s − 4·73-s + 12·79-s − 2·83-s + 2·85-s − 14·89-s − 2·95-s − 2·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.66·13-s + 0.485·17-s − 0.458·19-s + 1.25·23-s + 1/5·25-s + 1.11·29-s − 0.718·31-s − 1.31·37-s + 1.24·41-s − 0.152·43-s + 0.875·47-s − 49-s + 0.824·53-s + 0.520·59-s − 1.79·61-s − 0.744·65-s + 0.488·67-s − 0.949·71-s − 0.468·73-s + 1.35·79-s − 0.219·83-s + 0.216·85-s − 1.48·89-s − 0.205·95-s − 0.203·97-s + 0.0995·101-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 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−15.28378229801124, −14.75773340735733, −14.33344244441749, −13.92180632614447, −13.13770143960549, −12.74074367449530, −12.17537127248431, −11.82899697405605, −10.95865955493865, −10.48103448632707, −10.08419931990614, −9.268783800193478, −9.128841210404676, −8.268744544656148, −7.659808526170077, −7.051140205299876, −6.689382473364464, −5.810855004223408, −5.271810618740199, −4.762339779178217, −4.119550862914375, −3.134488683539288, −2.663115368826758, −1.927120399106757, −1.043679544610964, 0,
1.043679544610964, 1.927120399106757, 2.663115368826758, 3.134488683539288, 4.119550862914375, 4.762339779178217, 5.271810618740199, 5.810855004223408, 6.689382473364464, 7.051140205299876, 7.659808526170077, 8.268744544656148, 9.128841210404676, 9.268783800193478, 10.08419931990614, 10.48103448632707, 10.95865955493865, 11.82899697405605, 12.17537127248431, 12.74074367449530, 13.13770143960549, 13.92180632614447, 14.33344244441749, 14.75773340735733, 15.28378229801124
