| L(s) = 1 | − 3-s − 0.792·5-s − 3.80·7-s + 9-s − 5.89·11-s − 0.555·13-s + 0.792·15-s − 1.29·17-s + 0.877·19-s + 3.80·21-s − 1.89·23-s − 4.37·25-s − 27-s + 4.21·29-s − 6.06·31-s + 5.89·33-s + 3.01·35-s − 8.81·37-s + 0.555·39-s + 0.966·41-s − 1.48·43-s − 0.792·45-s − 4.12·47-s + 7.45·49-s + 1.29·51-s − 13.1·53-s + 4.66·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.354·5-s − 1.43·7-s + 0.333·9-s − 1.77·11-s − 0.154·13-s + 0.204·15-s − 0.313·17-s + 0.201·19-s + 0.829·21-s − 0.394·23-s − 0.874·25-s − 0.192·27-s + 0.782·29-s − 1.08·31-s + 1.02·33-s + 0.509·35-s − 1.44·37-s + 0.0889·39-s + 0.151·41-s − 0.226·43-s − 0.118·45-s − 0.601·47-s + 1.06·49-s + 0.180·51-s − 1.80·53-s + 0.629·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01373216811\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01373216811\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 0.792T + 5T^{2} \) |
| 7 | \( 1 + 3.80T + 7T^{2} \) |
| 11 | \( 1 + 5.89T + 11T^{2} \) |
| 13 | \( 1 + 0.555T + 13T^{2} \) |
| 17 | \( 1 + 1.29T + 17T^{2} \) |
| 19 | \( 1 - 0.877T + 19T^{2} \) |
| 23 | \( 1 + 1.89T + 23T^{2} \) |
| 29 | \( 1 - 4.21T + 29T^{2} \) |
| 31 | \( 1 + 6.06T + 31T^{2} \) |
| 37 | \( 1 + 8.81T + 37T^{2} \) |
| 41 | \( 1 - 0.966T + 41T^{2} \) |
| 43 | \( 1 + 1.48T + 43T^{2} \) |
| 47 | \( 1 + 4.12T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 + 7.38T + 61T^{2} \) |
| 67 | \( 1 - 6.38T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 + 0.414T + 79T^{2} \) |
| 83 | \( 1 - 2.57T + 83T^{2} \) |
| 89 | \( 1 - 0.384T + 89T^{2} \) |
| 97 | \( 1 + 3.26T + 97T^{2} \) |
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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
−7.70786910416393214034033140748, −7.15309527415192395109497991776, −6.39130259146502201083249221025, −5.79343598945988638634619888604, −5.12091686664944850534868966191, −4.36288714821197665838047027225, −3.37007085915841495906993131674, −2.87031154717512854089266876593, −1.77357158003425738843707968826, −0.05609283255498602513243611204,
0.05609283255498602513243611204, 1.77357158003425738843707968826, 2.87031154717512854089266876593, 3.37007085915841495906993131674, 4.36288714821197665838047027225, 5.12091686664944850534868966191, 5.79343598945988638634619888604, 6.39130259146502201083249221025, 7.15309527415192395109497991776, 7.70786910416393214034033140748
