| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 12-s − 2·13-s + 16-s − 18-s + 23-s − 24-s − 25-s + 2·26-s + 27-s − 32-s + 36-s − 2·39-s − 46-s − 2·47-s + 48-s − 49-s + 50-s − 2·52-s − 54-s − 2·59-s + 64-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 12-s − 2·13-s + 16-s − 18-s + 23-s − 24-s − 25-s + 2·26-s + 27-s − 32-s + 36-s − 2·39-s − 46-s − 2·47-s + 48-s − 49-s + 50-s − 2·52-s − 54-s − 2·59-s + 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6628601428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6628601428\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 + T )^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 29 | \( ( 1 - T )( 1 + T ) \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( ( 1 + T )^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( ( 1 + T )^{2} \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( ( 1 - T )^{2} \) |
| 73 | \( ( 1 - T )^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( ( 1 - T )( 1 + T ) \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−12.09302647396456597777346397286, −10.93946643197365168659828152374, −9.728827665828244985817206805142, −9.507900230546414812254038812813, −8.230744665147120095639978637918, −7.53732355803698750857256713166, −6.65570982609287744343710691672, −4.91736259196269300760071846596, −3.18395264755445911194883260299, −2.04338386115782083670441765174,
2.04338386115782083670441765174, 3.18395264755445911194883260299, 4.91736259196269300760071846596, 6.65570982609287744343710691672, 7.53732355803698750857256713166, 8.230744665147120095639978637918, 9.507900230546414812254038812813, 9.728827665828244985817206805142, 10.93946643197365168659828152374, 12.09302647396456597777346397286
