| L(s) = 1 | − 2-s − 3-s + 4-s − 0.343·5-s + 6-s + 2.39·7-s − 8-s + 9-s + 0.343·10-s + 1.78·11-s − 12-s + 13-s − 2.39·14-s + 0.343·15-s + 16-s + 5.22·17-s − 18-s − 3.94·19-s − 0.343·20-s − 2.39·21-s − 1.78·22-s + 2.02·23-s + 24-s − 4.88·25-s − 26-s − 27-s + 2.39·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.153·5-s + 0.408·6-s + 0.905·7-s − 0.353·8-s + 0.333·9-s + 0.108·10-s + 0.537·11-s − 0.288·12-s + 0.277·13-s − 0.640·14-s + 0.0885·15-s + 0.250·16-s + 1.26·17-s − 0.235·18-s − 0.905·19-s − 0.0767·20-s − 0.523·21-s − 0.379·22-s + 0.423·23-s + 0.204·24-s − 0.976·25-s − 0.196·26-s − 0.192·27-s + 0.452·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.362597841\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.362597841\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 + 0.343T + 5T^{2} \) |
| 7 | \( 1 - 2.39T + 7T^{2} \) |
| 11 | \( 1 - 1.78T + 11T^{2} \) |
| 17 | \( 1 - 5.22T + 17T^{2} \) |
| 19 | \( 1 + 3.94T + 19T^{2} \) |
| 23 | \( 1 - 2.02T + 23T^{2} \) |
| 29 | \( 1 + 7.94T + 29T^{2} \) |
| 31 | \( 1 - 2.26T + 31T^{2} \) |
| 37 | \( 1 - 8.14T + 37T^{2} \) |
| 41 | \( 1 + 0.158T + 41T^{2} \) |
| 43 | \( 1 - 2.52T + 43T^{2} \) |
| 47 | \( 1 - 7.07T + 47T^{2} \) |
| 53 | \( 1 - 6.10T + 53T^{2} \) |
| 59 | \( 1 + 8.74T + 59T^{2} \) |
| 61 | \( 1 - 3.19T + 61T^{2} \) |
| 67 | \( 1 - 7.84T + 67T^{2} \) |
| 71 | \( 1 - 7.09T + 71T^{2} \) |
| 73 | \( 1 - 17.0T + 73T^{2} \) |
| 79 | \( 1 + 3.08T + 79T^{2} \) |
| 83 | \( 1 + 9.65T + 83T^{2} \) |
| 89 | \( 1 - 6.89T + 89T^{2} \) |
| 97 | \( 1 - 7.83T + 97T^{2} \) |
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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
−7.85569861008566060940843725281, −7.32117110717797596217610412364, −6.46070050858439558206955861745, −5.82403893197717985714058663478, −5.20374131983726350364225068423, −4.23002944880578177590308185026, −3.63349135513609992653832056222, −2.38139304631469384099528664002, −1.53885317331267813843548413787, −0.70820018753726197319537706233,
0.70820018753726197319537706233, 1.53885317331267813843548413787, 2.38139304631469384099528664002, 3.63349135513609992653832056222, 4.23002944880578177590308185026, 5.20374131983726350364225068423, 5.82403893197717985714058663478, 6.46070050858439558206955861745, 7.32117110717797596217610412364, 7.85569861008566060940843725281
