| L(s) = 1 | − 2-s − 3-s + 4-s + 2.28·5-s + 6-s − 1.19·7-s − 8-s + 9-s − 2.28·10-s + 4.27·11-s − 12-s + 13-s + 1.19·14-s − 2.28·15-s + 16-s − 3.59·17-s − 18-s − 1.35·19-s + 2.28·20-s + 1.19·21-s − 4.27·22-s − 6.42·23-s + 24-s + 0.241·25-s − 26-s − 27-s − 1.19·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.02·5-s + 0.408·6-s − 0.451·7-s − 0.353·8-s + 0.333·9-s − 0.723·10-s + 1.29·11-s − 0.288·12-s + 0.277·13-s + 0.319·14-s − 0.591·15-s + 0.250·16-s − 0.872·17-s − 0.235·18-s − 0.311·19-s + 0.511·20-s + 0.260·21-s − 0.912·22-s − 1.33·23-s + 0.204·24-s + 0.0483·25-s − 0.196·26-s − 0.192·27-s − 0.225·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.228739397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.228739397\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 - 2.28T + 5T^{2} \) |
| 7 | \( 1 + 1.19T + 7T^{2} \) |
| 11 | \( 1 - 4.27T + 11T^{2} \) |
| 17 | \( 1 + 3.59T + 17T^{2} \) |
| 19 | \( 1 + 1.35T + 19T^{2} \) |
| 23 | \( 1 + 6.42T + 23T^{2} \) |
| 29 | \( 1 + 6.53T + 29T^{2} \) |
| 31 | \( 1 + 1.25T + 31T^{2} \) |
| 37 | \( 1 + 6.96T + 37T^{2} \) |
| 41 | \( 1 - 7.36T + 41T^{2} \) |
| 43 | \( 1 - 1.70T + 43T^{2} \) |
| 47 | \( 1 - 1.19T + 47T^{2} \) |
| 53 | \( 1 - 2.84T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 - 3.12T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 - 1.20T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 + 2.00T + 79T^{2} \) |
| 83 | \( 1 - 4.49T + 83T^{2} \) |
| 89 | \( 1 - 5.55T + 89T^{2} \) |
| 97 | \( 1 - 7.35T + 97T^{2} \) |
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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
−7.84509940883857872683386228669, −6.89515879861318731309937648239, −6.52049655421087663121516140308, −5.92987964955260696976905348554, −5.35174475465480757039955474948, −4.15540767621612142369487368984, −3.60779150013175953675925748948, −2.19814247506004274765773294742, −1.80055470522176618120902852368, −0.62445002108060695948292282719,
0.62445002108060695948292282719, 1.80055470522176618120902852368, 2.19814247506004274765773294742, 3.60779150013175953675925748948, 4.15540767621612142369487368984, 5.35174475465480757039955474948, 5.92987964955260696976905348554, 6.52049655421087663121516140308, 6.89515879861318731309937648239, 7.84509940883857872683386228669
