| L(s) = 1 | − 3-s + 0.0581·5-s − 2.78·7-s + 9-s + 3.35·11-s − 1.13·13-s − 0.0581·15-s + 1.84·17-s − 0.820·19-s + 2.78·21-s − 4.45·23-s − 4.99·25-s − 27-s + 2.16·29-s − 4.12·31-s − 3.35·33-s − 0.161·35-s + 2.83·37-s + 1.13·39-s + 11.8·41-s − 2.14·43-s + 0.0581·45-s + 2.54·47-s + 0.745·49-s − 1.84·51-s + 9.14·53-s + 0.195·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.0259·5-s − 1.05·7-s + 0.333·9-s + 1.01·11-s − 0.316·13-s − 0.0150·15-s + 0.447·17-s − 0.188·19-s + 0.607·21-s − 0.929·23-s − 0.999·25-s − 0.192·27-s + 0.401·29-s − 0.740·31-s − 0.584·33-s − 0.0273·35-s + 0.465·37-s + 0.182·39-s + 1.85·41-s − 0.326·43-s + 0.00866·45-s + 0.371·47-s + 0.106·49-s − 0.258·51-s + 1.25·53-s + 0.0263·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.153704994\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.153704994\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 503 | \( 1 + T \) |
| good | 5 | \( 1 - 0.0581T + 5T^{2} \) |
| 7 | \( 1 + 2.78T + 7T^{2} \) |
| 11 | \( 1 - 3.35T + 11T^{2} \) |
| 13 | \( 1 + 1.13T + 13T^{2} \) |
| 17 | \( 1 - 1.84T + 17T^{2} \) |
| 19 | \( 1 + 0.820T + 19T^{2} \) |
| 23 | \( 1 + 4.45T + 23T^{2} \) |
| 29 | \( 1 - 2.16T + 29T^{2} \) |
| 31 | \( 1 + 4.12T + 31T^{2} \) |
| 37 | \( 1 - 2.83T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 + 2.14T + 43T^{2} \) |
| 47 | \( 1 - 2.54T + 47T^{2} \) |
| 53 | \( 1 - 9.14T + 53T^{2} \) |
| 59 | \( 1 + 5.43T + 59T^{2} \) |
| 61 | \( 1 + 3.80T + 61T^{2} \) |
| 67 | \( 1 + 8.54T + 67T^{2} \) |
| 71 | \( 1 - 7.66T + 71T^{2} \) |
| 73 | \( 1 - 6.97T + 73T^{2} \) |
| 79 | \( 1 - 3.92T + 79T^{2} \) |
| 83 | \( 1 + 7.00T + 83T^{2} \) |
| 89 | \( 1 + 0.190T + 89T^{2} \) |
| 97 | \( 1 + 5.39T + 97T^{2} \) |
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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
−7.919407638442506951583073500571, −7.30423973325718076760557733814, −6.48099804983760706165998259795, −6.06336505860867497931295167652, −5.39832916075589557651562443045, −4.25652550087061325565099291555, −3.83016681977890720995791511952, −2.82202567463378342340885637746, −1.76606932584645474735627190650, −0.57799063690462197429439185325,
0.57799063690462197429439185325, 1.76606932584645474735627190650, 2.82202567463378342340885637746, 3.83016681977890720995791511952, 4.25652550087061325565099291555, 5.39832916075589557651562443045, 6.06336505860867497931295167652, 6.48099804983760706165998259795, 7.30423973325718076760557733814, 7.919407638442506951583073500571
