| L(s) = 1 | + 3-s − 3.35·5-s + 2.96·7-s + 9-s + 1.61·11-s − 2·13-s − 3.35·15-s + 4.96·17-s − 1.35·19-s + 2.96·21-s + 23-s + 6.22·25-s + 27-s + 7.92·29-s − 5.92·31-s + 1.61·33-s − 9.92·35-s + 2.31·37-s − 2·39-s − 1.22·41-s − 4.57·43-s − 3.35·45-s − 1.92·47-s + 1.77·49-s + 4.96·51-s + 4.12·53-s − 5.40·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.49·5-s + 1.11·7-s + 0.333·9-s + 0.486·11-s − 0.554·13-s − 0.865·15-s + 1.20·17-s − 0.309·19-s + 0.646·21-s + 0.208·23-s + 1.24·25-s + 0.192·27-s + 1.47·29-s − 1.06·31-s + 0.280·33-s − 1.67·35-s + 0.380·37-s − 0.320·39-s − 0.191·41-s − 0.697·43-s − 0.499·45-s − 0.280·47-s + 0.253·49-s + 0.694·51-s + 0.566·53-s − 0.728·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.112202543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.112202543\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 3.35T + 5T^{2} \) |
| 7 | \( 1 - 2.96T + 7T^{2} \) |
| 11 | \( 1 - 1.61T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 4.96T + 17T^{2} \) |
| 19 | \( 1 + 1.35T + 19T^{2} \) |
| 29 | \( 1 - 7.92T + 29T^{2} \) |
| 31 | \( 1 + 5.92T + 31T^{2} \) |
| 37 | \( 1 - 2.31T + 37T^{2} \) |
| 41 | \( 1 + 1.22T + 41T^{2} \) |
| 43 | \( 1 + 4.57T + 43T^{2} \) |
| 47 | \( 1 + 1.92T + 47T^{2} \) |
| 53 | \( 1 - 4.12T + 53T^{2} \) |
| 59 | \( 1 - 2.70T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 - 4.57T + 67T^{2} \) |
| 71 | \( 1 - 16.6T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 - 5.03T + 79T^{2} \) |
| 83 | \( 1 + 15.0T + 83T^{2} \) |
| 89 | \( 1 + 5.73T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−8.369896947383956966848539841167, −7.58951273879093849695892393675, −7.34392379727034108292487532656, −6.29666805689443437097105519520, −5.06766370451500861732498061167, −4.59113175451670621091919862042, −3.74795794869359279904455040671, −3.11854852339768771460512680952, −1.92332463200877109556400780449, −0.818388173607787936613950388616,
0.818388173607787936613950388616, 1.92332463200877109556400780449, 3.11854852339768771460512680952, 3.74795794869359279904455040671, 4.59113175451670621091919862042, 5.06766370451500861732498061167, 6.29666805689443437097105519520, 7.34392379727034108292487532656, 7.58951273879093849695892393675, 8.369896947383956966848539841167
