| L(s) = 1 | + 1.41·3-s − 1.41·5-s − 0.999·9-s − 11-s − 2.00·15-s + 5.65·17-s + 6·23-s − 2.99·25-s − 5.65·27-s − 6·29-s − 7.07·31-s − 1.41·33-s + 6·37-s + 8·43-s + 1.41·45-s − 1.41·47-s + 8.00·51-s + 1.41·55-s − 9.89·59-s + 8.48·61-s − 14·67-s + 8.48·69-s − 2·71-s − 2.82·73-s − 4.24·75-s − 5.00·81-s + 5.65·83-s + ⋯ |
| L(s) = 1 | + 0.816·3-s − 0.632·5-s − 0.333·9-s − 0.301·11-s − 0.516·15-s + 1.37·17-s + 1.25·23-s − 0.599·25-s − 1.08·27-s − 1.11·29-s − 1.27·31-s − 0.246·33-s + 0.986·37-s + 1.21·43-s + 0.210·45-s − 0.206·47-s + 1.12·51-s + 0.190·55-s − 1.28·59-s + 1.08·61-s − 1.71·67-s + 1.02·69-s − 0.237·71-s − 0.331·73-s − 0.489·75-s − 0.555·81-s + 0.620·83-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 | \( 1 + 1.41T + 5T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 5.65T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 7.07T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 1.41T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 9.89T + 59T^{2} \) |
| 61 | \( 1 - 8.48T + 61T^{2} \) |
| 67 | \( 1 + 14T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + 2.82T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 5.65T + 83T^{2} \) |
| 89 | \( 1 + 18.3T + 89T^{2} \) |
| 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
−7.73305369387117428712672219993, −7.03498009342415921998020612849, −5.83664779659584769093522607043, −5.51017997632402956614111360012, −4.47020509907062789552691365190, −3.66393956060997362476914729952, −3.16287836109502122581723470269, −2.38630100965440083879586096079, −1.30648916402694460465835732440, 0,
1.30648916402694460465835732440, 2.38630100965440083879586096079, 3.16287836109502122581723470269, 3.66393956060997362476914729952, 4.47020509907062789552691365190, 5.51017997632402956614111360012, 5.83664779659584769093522607043, 7.03498009342415921998020612849, 7.73305369387117428712672219993
