| L(s) = 1 | + 2-s + 3-s + 4-s + 2.64·5-s + 6-s + 4.97·7-s + 8-s + 9-s + 2.64·10-s + 0.421·11-s + 12-s + 13-s + 4.97·14-s + 2.64·15-s + 16-s − 5.90·17-s + 18-s + 3.97·19-s + 2.64·20-s + 4.97·21-s + 0.421·22-s − 3.99·23-s + 24-s + 1.99·25-s + 26-s + 27-s + 4.97·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.18·5-s + 0.408·6-s + 1.88·7-s + 0.353·8-s + 0.333·9-s + 0.836·10-s + 0.127·11-s + 0.288·12-s + 0.277·13-s + 1.33·14-s + 0.682·15-s + 0.250·16-s − 1.43·17-s + 0.235·18-s + 0.912·19-s + 0.591·20-s + 1.08·21-s + 0.0899·22-s − 0.833·23-s + 0.204·24-s + 0.398·25-s + 0.196·26-s + 0.192·27-s + 0.940·28-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
| \(L(1)\) |
\(\approx\) |
\(6.926497265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.926497265\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.64T + 5T^{2} \) |
| 7 | \( 1 - 4.97T + 7T^{2} \) |
| 11 | \( 1 - 0.421T + 11T^{2} \) |
| 17 | \( 1 + 5.90T + 17T^{2} \) |
| 19 | \( 1 - 3.97T + 19T^{2} \) |
| 23 | \( 1 + 3.99T + 23T^{2} \) |
| 29 | \( 1 - 8.29T + 29T^{2} \) |
| 31 | \( 1 + 3.97T + 31T^{2} \) |
| 37 | \( 1 + 1.74T + 37T^{2} \) |
| 41 | \( 1 + 8.23T + 41T^{2} \) |
| 43 | \( 1 + 1.01T + 43T^{2} \) |
| 47 | \( 1 - 1.18T + 47T^{2} \) |
| 53 | \( 1 - 4.35T + 53T^{2} \) |
| 59 | \( 1 - 0.467T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 + 6.73T + 67T^{2} \) |
| 71 | \( 1 - 4.54T + 71T^{2} \) |
| 73 | \( 1 + 5.66T + 73T^{2} \) |
| 79 | \( 1 - 3.58T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 - 1.28T + 89T^{2} \) |
| 97 | \( 1 + 1.27T + 97T^{2} \) |
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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
−7.82125210939281136224895086766, −7.11562658814120456149039725729, −6.33449992824564376400601480446, −5.65205521443120702301672771373, −4.88658612442490341209591311922, −4.51524242514031547617072194709, −3.55364482707123491102136673553, −2.48211759515138022517340585796, −1.91810035279241648192200165134, −1.32001503075844620774958763263,
1.32001503075844620774958763263, 1.91810035279241648192200165134, 2.48211759515138022517340585796, 3.55364482707123491102136673553, 4.51524242514031547617072194709, 4.88658612442490341209591311922, 5.65205521443120702301672771373, 6.33449992824564376400601480446, 7.11562658814120456149039725729, 7.82125210939281136224895086766
