| L(s) = 1 | + 2·3-s + 4·5-s + 7-s + 9-s + 8·15-s + 2·17-s + 2·19-s + 2·21-s − 8·23-s + 11·25-s − 4·27-s − 2·29-s − 4·31-s + 4·35-s − 2·41-s − 8·43-s + 4·45-s − 4·47-s + 49-s + 4·51-s − 10·53-s + 4·57-s − 6·59-s − 4·61-s + 63-s − 12·67-s − 16·69-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.78·5-s + 0.377·7-s + 1/3·9-s + 2.06·15-s + 0.485·17-s + 0.458·19-s + 0.436·21-s − 1.66·23-s + 11/5·25-s − 0.769·27-s − 0.371·29-s − 0.718·31-s + 0.676·35-s − 0.312·41-s − 1.21·43-s + 0.596·45-s − 0.583·47-s + 1/7·49-s + 0.560·51-s − 1.37·53-s + 0.529·57-s − 0.781·59-s − 0.512·61-s + 0.125·63-s − 1.46·67-s − 1.92·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−14.35801527375651, −13.73691923742483, −13.51472504377908, −13.07969604125005, −12.40725035058110, −11.87638824146435, −11.26878919966318, −10.52462661375774, −10.17143743000642, −9.627143070018867, −9.309255732420704, −8.835430254452259, −8.211219474016051, −7.788978852493099, −7.232784510593116, −6.404562834456506, −6.013867500952746, −5.487400937327903, −4.953425211695578, −4.223938782151136, −3.388799988990764, −3.004312195301771, −2.277865305321467, −1.700273891404453, −1.464899323184333, 0,
1.464899323184333, 1.700273891404453, 2.277865305321467, 3.004312195301771, 3.388799988990764, 4.223938782151136, 4.953425211695578, 5.487400937327903, 6.013867500952746, 6.404562834456506, 7.232784510593116, 7.788978852493099, 8.211219474016051, 8.835430254452259, 9.309255732420704, 9.627143070018867, 10.17143743000642, 10.52462661375774, 11.26878919966318, 11.87638824146435, 12.40725035058110, 13.07969604125005, 13.51472504377908, 13.73691923742483, 14.35801527375651
