| L(s) = 1 | − 3-s − 2.26·5-s + 2.39·7-s + 9-s + 3.59·11-s − 4.12·13-s + 2.26·15-s − 4.77·17-s − 2.73·19-s − 2.39·21-s + 4.11·23-s + 0.136·25-s − 27-s − 6.23·29-s + 9.50·31-s − 3.59·33-s − 5.43·35-s − 5.59·37-s + 4.12·39-s + 7.03·41-s + 2.43·43-s − 2.26·45-s + 13.0·47-s − 1.24·49-s + 4.77·51-s + 4.09·53-s − 8.15·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.01·5-s + 0.906·7-s + 0.333·9-s + 1.08·11-s − 1.14·13-s + 0.585·15-s − 1.15·17-s − 0.627·19-s − 0.523·21-s + 0.857·23-s + 0.0273·25-s − 0.192·27-s − 1.15·29-s + 1.70·31-s − 0.626·33-s − 0.919·35-s − 0.919·37-s + 0.660·39-s + 1.09·41-s + 0.372·43-s − 0.337·45-s + 1.90·47-s − 0.177·49-s + 0.668·51-s + 0.562·53-s − 1.10·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.150232303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.150232303\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 251 | \( 1 - T \) |
| good | 5 | \( 1 + 2.26T + 5T^{2} \) |
| 7 | \( 1 - 2.39T + 7T^{2} \) |
| 11 | \( 1 - 3.59T + 11T^{2} \) |
| 13 | \( 1 + 4.12T + 13T^{2} \) |
| 17 | \( 1 + 4.77T + 17T^{2} \) |
| 19 | \( 1 + 2.73T + 19T^{2} \) |
| 23 | \( 1 - 4.11T + 23T^{2} \) |
| 29 | \( 1 + 6.23T + 29T^{2} \) |
| 31 | \( 1 - 9.50T + 31T^{2} \) |
| 37 | \( 1 + 5.59T + 37T^{2} \) |
| 41 | \( 1 - 7.03T + 41T^{2} \) |
| 43 | \( 1 - 2.43T + 43T^{2} \) |
| 47 | \( 1 - 13.0T + 47T^{2} \) |
| 53 | \( 1 - 4.09T + 53T^{2} \) |
| 59 | \( 1 + 0.223T + 59T^{2} \) |
| 61 | \( 1 + 1.56T + 61T^{2} \) |
| 67 | \( 1 - 0.136T + 67T^{2} \) |
| 71 | \( 1 + 12.9T + 71T^{2} \) |
| 73 | \( 1 - 2.01T + 73T^{2} \) |
| 79 | \( 1 + 5.29T + 79T^{2} \) |
| 83 | \( 1 + 6.10T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 + 3.78T + 97T^{2} \) |
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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
−7.957698583156155697749177899548, −7.30891442808558867055874026696, −6.82492287626667074535716839749, −5.95868450635424147590214094936, −5.05225679441389408451428798338, −4.31338330118660626996412948789, −4.07564135439693923157665319425, −2.72999876420607809542676755492, −1.75053640564796177391145903787, −0.58418461016557358415988249810,
0.58418461016557358415988249810, 1.75053640564796177391145903787, 2.72999876420607809542676755492, 4.07564135439693923157665319425, 4.31338330118660626996412948789, 5.05225679441389408451428798338, 5.95868450635424147590214094936, 6.82492287626667074535716839749, 7.30891442808558867055874026696, 7.957698583156155697749177899548
