| L(s) = 1 | − 0.371·2-s − 1.74·3-s − 1.86·4-s + 5-s + 0.647·6-s + 7-s + 1.43·8-s + 0.0278·9-s − 0.371·10-s + 3.23·12-s − 5.54·13-s − 0.371·14-s − 1.74·15-s + 3.18·16-s − 6.07·17-s − 0.0103·18-s + 0.160·19-s − 1.86·20-s − 1.74·21-s − 5.95·23-s − 2.49·24-s + 25-s + 2.06·26-s + 5.17·27-s − 1.86·28-s + 5.59·29-s + 0.647·30-s + ⋯ |
| L(s) = 1 | − 0.263·2-s − 1.00·3-s − 0.930·4-s + 0.447·5-s + 0.264·6-s + 0.377·7-s + 0.507·8-s + 0.00927·9-s − 0.117·10-s + 0.935·12-s − 1.53·13-s − 0.0994·14-s − 0.449·15-s + 0.797·16-s − 1.47·17-s − 0.00243·18-s + 0.0367·19-s − 0.416·20-s − 0.379·21-s − 1.24·23-s − 0.510·24-s + 0.200·25-s + 0.404·26-s + 0.995·27-s − 0.351·28-s + 1.03·29-s + 0.118·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4089255859\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4089255859\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.371T + 2T^{2} \) |
| 3 | \( 1 + 1.74T + 3T^{2} \) |
| 13 | \( 1 + 5.54T + 13T^{2} \) |
| 17 | \( 1 + 6.07T + 17T^{2} \) |
| 19 | \( 1 - 0.160T + 19T^{2} \) |
| 23 | \( 1 + 5.95T + 23T^{2} \) |
| 29 | \( 1 - 5.59T + 29T^{2} \) |
| 31 | \( 1 + 3.38T + 31T^{2} \) |
| 37 | \( 1 + 5.98T + 37T^{2} \) |
| 41 | \( 1 - 7.71T + 41T^{2} \) |
| 43 | \( 1 - 2.74T + 43T^{2} \) |
| 47 | \( 1 - 0.838T + 47T^{2} \) |
| 53 | \( 1 + 9.90T + 53T^{2} \) |
| 59 | \( 1 - 5.27T + 59T^{2} \) |
| 61 | \( 1 + 5.19T + 61T^{2} \) |
| 67 | \( 1 - 5.66T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + 9.17T + 79T^{2} \) |
| 83 | \( 1 - 5.09T + 83T^{2} \) |
| 89 | \( 1 - 2.46T + 89T^{2} \) |
| 97 | \( 1 - 11.9T + 97T^{2} \) |
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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
−8.510108525786223214396644095231, −7.68495523787158275122532524447, −6.90636320212624565038946844558, −6.05088728589255712418138381449, −5.39638955420327620086398421203, −4.69111993574105396098800594858, −4.25970354654808859678797084788, −2.79223781159816943225780821410, −1.77659607682595808337630705541, −0.39109897555852846520368687803,
0.39109897555852846520368687803, 1.77659607682595808337630705541, 2.79223781159816943225780821410, 4.25970354654808859678797084788, 4.69111993574105396098800594858, 5.39638955420327620086398421203, 6.05088728589255712418138381449, 6.90636320212624565038946844558, 7.68495523787158275122532524447, 8.510108525786223214396644095231
