| L(s) = 1 | − 3-s + 4·5-s − 4·7-s − 2·9-s + 5·11-s − 4·13-s − 4·15-s − 2·17-s + 4·21-s + 11·25-s + 5·27-s + 4·29-s + 8·31-s − 5·33-s − 16·35-s + 8·37-s + 4·39-s − 3·41-s + 8·43-s − 8·45-s − 12·47-s + 9·49-s + 2·51-s + 12·53-s + 20·55-s − 3·59-s − 4·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.78·5-s − 1.51·7-s − 2/3·9-s + 1.50·11-s − 1.10·13-s − 1.03·15-s − 0.485·17-s + 0.872·21-s + 11/5·25-s + 0.962·27-s + 0.742·29-s + 1.43·31-s − 0.870·33-s − 2.70·35-s + 1.31·37-s + 0.640·39-s − 0.468·41-s + 1.21·43-s − 1.19·45-s − 1.75·47-s + 9/7·49-s + 0.280·51-s + 1.64·53-s + 2.69·55-s − 0.390·59-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.385975379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.385975379\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−13.95884298457293, −13.34683167493448, −12.88804882049654, −12.34373645905249, −11.98962795867386, −11.44795841045645, −10.77496793351507, −10.21673879055202, −9.725363185318973, −9.612149800859130, −8.966526096780833, −8.651143167982105, −7.704948655524951, −6.828649633686114, −6.481912942908690, −6.317213902062261, −5.838002607088653, −5.137232813667236, −4.670671130924764, −3.910967676574481, −3.033653666816243, −2.655576676736166, −2.108967982250493, −1.143475283284770, −0.5543487880617536,
0.5543487880617536, 1.143475283284770, 2.108967982250493, 2.655576676736166, 3.033653666816243, 3.910967676574481, 4.670671130924764, 5.137232813667236, 5.838002607088653, 6.317213902062261, 6.481912942908690, 6.828649633686114, 7.704948655524951, 8.651143167982105, 8.966526096780833, 9.612149800859130, 9.725363185318973, 10.21673879055202, 10.77496793351507, 11.44795841045645, 11.98962795867386, 12.34373645905249, 12.88804882049654, 13.34683167493448, 13.95884298457293
