| L(s) = 1 | − 5-s + 1.44·7-s − 2·17-s − 2.89·19-s + 2.55·23-s + 25-s + 7.89·29-s − 10.8·31-s − 1.44·35-s − 6·37-s − 0.101·41-s + 7.79·43-s + 4.55·47-s − 4.89·49-s − 11.7·53-s + 10.8·59-s − 3·61-s − 11.2·67-s + 9.79·71-s − 5.79·73-s − 2.89·79-s + 0.550·83-s + 2·85-s − 16.7·89-s + 2.89·95-s + 2·97-s − 2·101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.547·7-s − 0.485·17-s − 0.665·19-s + 0.531·23-s + 0.200·25-s + 1.46·29-s − 1.95·31-s − 0.245·35-s − 0.986·37-s − 0.0157·41-s + 1.18·43-s + 0.663·47-s − 0.699·49-s − 1.62·53-s + 1.41·59-s − 0.384·61-s − 1.37·67-s + 1.16·71-s − 0.678·73-s − 0.326·79-s + 0.0604·83-s + 0.216·85-s − 1.78·89-s + 0.297·95-s + 0.203·97-s − 0.199·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| good | 7 | \( 1 - 1.44T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 2.89T + 19T^{2} \) |
| 23 | \( 1 - 2.55T + 23T^{2} \) |
| 29 | \( 1 - 7.89T + 29T^{2} \) |
| 31 | \( 1 + 10.8T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 0.101T + 41T^{2} \) |
| 43 | \( 1 - 7.79T + 43T^{2} \) |
| 47 | \( 1 - 4.55T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + 3T + 61T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 - 9.79T + 71T^{2} \) |
| 73 | \( 1 + 5.79T + 73T^{2} \) |
| 79 | \( 1 + 2.89T + 79T^{2} \) |
| 83 | \( 1 - 0.550T + 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−7.61259979343361470049431124958, −7.05670769903213810545746249503, −6.32061538772238072790485298279, −5.45191405024685199131736079253, −4.72597201617999733469843463947, −4.09740363913011287335324522534, −3.22156468883634681010563923872, −2.28369580479480489476430218731, −1.32234569409703140399230656624, 0,
1.32234569409703140399230656624, 2.28369580479480489476430218731, 3.22156468883634681010563923872, 4.09740363913011287335324522534, 4.72597201617999733469843463947, 5.45191405024685199131736079253, 6.32061538772238072790485298279, 7.05670769903213810545746249503, 7.61259979343361470049431124958
