| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 4·5-s − 2·6-s + 9-s − 8·10-s − 5·11-s − 2·12-s + 2·13-s + 4·15-s − 4·16-s + 2·18-s + 5·19-s − 8·20-s − 10·22-s − 23-s + 11·25-s + 4·26-s − 27-s − 2·29-s + 8·30-s − 6·31-s − 8·32-s + 5·33-s + 2·36-s + 6·37-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 1.78·5-s − 0.816·6-s + 1/3·9-s − 2.52·10-s − 1.50·11-s − 0.577·12-s + 0.554·13-s + 1.03·15-s − 16-s + 0.471·18-s + 1.14·19-s − 1.78·20-s − 2.13·22-s − 0.208·23-s + 11/5·25-s + 0.784·26-s − 0.192·27-s − 0.371·29-s + 1.46·30-s − 1.07·31-s − 1.41·32-s + 0.870·33-s + 1/3·36-s + 0.986·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.561018265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.561018265\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−8.355628510736615990636922865398, −7.52808643864842167704210656420, −7.21599931075485283729377776267, −6.06608389103752477000738441939, −5.36867373932747341880977289652, −4.79367865344036019029087978497, −3.94905053159347170606487756755, −3.45361589111826740147222343405, −2.51112652126840295893813807038, −0.59317848488426051004393238239,
0.59317848488426051004393238239, 2.51112652126840295893813807038, 3.45361589111826740147222343405, 3.94905053159347170606487756755, 4.79367865344036019029087978497, 5.36867373932747341880977289652, 6.06608389103752477000738441939, 7.21599931075485283729377776267, 7.52808643864842167704210656420, 8.355628510736615990636922865398
