| L(s) = 1 | − 0.738·3-s − 3.70·5-s − 1.43·7-s − 2.45·9-s − 2.99·11-s − 13-s + 2.73·15-s + 2.23·17-s − 1.24·19-s + 1.05·21-s − 8.24·23-s + 8.75·25-s + 4.02·27-s − 10.2·29-s + 10.4·31-s + 2.21·33-s + 5.30·35-s − 6.93·37-s + 0.738·39-s + 1.17·41-s + 9.90·43-s + 9.10·45-s + 9.11·47-s − 4.95·49-s − 1.64·51-s − 2.82·53-s + 11.1·55-s + ⋯ |
| L(s) = 1 | − 0.426·3-s − 1.65·5-s − 0.541·7-s − 0.818·9-s − 0.902·11-s − 0.277·13-s + 0.707·15-s + 0.541·17-s − 0.286·19-s + 0.230·21-s − 1.71·23-s + 1.75·25-s + 0.775·27-s − 1.90·29-s + 1.88·31-s + 0.384·33-s + 0.897·35-s − 1.13·37-s + 0.118·39-s + 0.183·41-s + 1.51·43-s + 1.35·45-s + 1.33·47-s − 0.707·49-s − 0.230·51-s − 0.387·53-s + 1.49·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3724030367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3724030367\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 0.738T + 3T^{2} \) |
| 5 | \( 1 + 3.70T + 5T^{2} \) |
| 7 | \( 1 + 1.43T + 7T^{2} \) |
| 11 | \( 1 + 2.99T + 11T^{2} \) |
| 17 | \( 1 - 2.23T + 17T^{2} \) |
| 19 | \( 1 + 1.24T + 19T^{2} \) |
| 23 | \( 1 + 8.24T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 + 6.93T + 37T^{2} \) |
| 41 | \( 1 - 1.17T + 41T^{2} \) |
| 43 | \( 1 - 9.90T + 43T^{2} \) |
| 47 | \( 1 - 9.11T + 47T^{2} \) |
| 53 | \( 1 + 2.82T + 53T^{2} \) |
| 59 | \( 1 + 2.99T + 59T^{2} \) |
| 61 | \( 1 - 1.77T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 - 3.08T + 71T^{2} \) |
| 73 | \( 1 - 1.90T + 73T^{2} \) |
| 79 | \( 1 + 1.98T + 79T^{2} \) |
| 83 | \( 1 + 2.21T + 83T^{2} \) |
| 89 | \( 1 - 6.33T + 89T^{2} \) |
| 97 | \( 1 + 14.5T + 97T^{2} \) |
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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
−9.347096226141872199776928850435, −8.240778612680678146488046262619, −7.934484402137949236679301376430, −7.09246393525964314515865752750, −6.07130259585721338972298712303, −5.30542143549857746428752359672, −4.25161319890896993567102855671, −3.49628878158945217648480905639, −2.51367978789331340744975022305, −0.40430143103612874809424454455,
0.40430143103612874809424454455, 2.51367978789331340744975022305, 3.49628878158945217648480905639, 4.25161319890896993567102855671, 5.30542143549857746428752359672, 6.07130259585721338972298712303, 7.09246393525964314515865752750, 7.934484402137949236679301376430, 8.240778612680678146488046262619, 9.347096226141872199776928850435
