| L(s) = 1 | − 1.41·3-s + 4.24·5-s − 0.999·9-s + 11-s − 6·15-s − 5.65·17-s − 6·23-s + 12.9·25-s + 5.65·27-s + 2·29-s + 1.41·31-s − 1.41·33-s − 10·37-s − 11.3·41-s + 8·43-s − 4.24·45-s − 4.24·47-s + 8.00·51-s + 8·53-s + 4.24·55-s − 1.41·59-s − 2.82·61-s − 2·67-s + 8.48·69-s + 2·71-s + 8.48·73-s − 18.3·75-s + ⋯ |
| L(s) = 1 | − 0.816·3-s + 1.89·5-s − 0.333·9-s + 0.301·11-s − 1.54·15-s − 1.37·17-s − 1.25·23-s + 2.59·25-s + 1.08·27-s + 0.371·29-s + 0.254·31-s − 0.246·33-s − 1.64·37-s − 1.76·41-s + 1.21·43-s − 0.632·45-s − 0.618·47-s + 1.12·51-s + 1.09·53-s + 0.572·55-s − 0.184·59-s − 0.362·61-s − 0.244·67-s + 1.02·69-s + 0.237·71-s + 0.993·73-s − 2.12·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 - 4.24T + 5T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 5.65T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 4.24T + 47T^{2} \) |
| 53 | \( 1 - 8T + 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 + 2.82T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 8.48T + 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 - 16.9T + 83T^{2} \) |
| 89 | \( 1 + 7.07T + 89T^{2} \) |
| 97 | \( 1 + 9.89T + 97T^{2} \) |
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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
−7.04986369267803997510165343985, −6.48403834581503661527856868870, −6.14910548389870848672739741291, −5.36464905287679354488287633663, −4.99556651167802429896464990057, −3.99971803263474356635919120135, −2.81201898549579342942276497210, −2.12059141167621194980267202069, −1.35801922528529719030194429191, 0,
1.35801922528529719030194429191, 2.12059141167621194980267202069, 2.81201898549579342942276497210, 3.99971803263474356635919120135, 4.99556651167802429896464990057, 5.36464905287679354488287633663, 6.14910548389870848672739741291, 6.48403834581503661527856868870, 7.04986369267803997510165343985
