| L(s) = 1 | + 2-s − 1.87·3-s + 4-s − 1.87·6-s + 1.53·7-s + 8-s + 2.53·9-s − 1.87·12-s + 0.347·13-s + 1.53·14-s + 16-s + 0.347·17-s + 2.53·18-s + 19-s − 2.87·21-s − 1.87·23-s − 1.87·24-s + 0.347·26-s − 2.87·27-s + 1.53·28-s + 0.347·29-s + 32-s + 0.347·34-s + 2.53·36-s − 37-s + 38-s − 0.652·39-s + ⋯ |
| L(s) = 1 | + 2-s − 1.87·3-s + 4-s − 1.87·6-s + 1.53·7-s + 8-s + 2.53·9-s − 1.87·12-s + 0.347·13-s + 1.53·14-s + 16-s + 0.347·17-s + 2.53·18-s + 19-s − 2.87·21-s − 1.87·23-s − 1.87·24-s + 0.347·26-s − 2.87·27-s + 1.53·28-s + 0.347·29-s + 32-s + 0.347·34-s + 2.53·36-s − 37-s + 38-s − 0.652·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.769037272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.769037272\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 1.87T + T^{2} \) |
| 7 | \( 1 - 1.53T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - 0.347T + T^{2} \) |
| 17 | \( 1 - 0.347T + T^{2} \) |
| 23 | \( 1 + 1.87T + T^{2} \) |
| 29 | \( 1 - 0.347T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + T + T^{2} \) |
| 53 | \( 1 - 1.53T + T^{2} \) |
| 59 | \( 1 + 1.87T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.53T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 1.87T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.351353760629594578274853851324, −7.60832019804577456202142826120, −7.02678264425828841444314941909, −6.06449188826151788193384752646, −5.66455582646833053538356700807, −4.95200572745320401436093592935, −4.47188360610106782843205787481, −3.60587576232893532899609522099, −1.95667818261153896396445420352, −1.22480060681873618762671818129,
1.22480060681873618762671818129, 1.95667818261153896396445420352, 3.60587576232893532899609522099, 4.47188360610106782843205787481, 4.95200572745320401436093592935, 5.66455582646833053538356700807, 6.06449188826151788193384752646, 7.02678264425828841444314941909, 7.60832019804577456202142826120, 8.351353760629594578274853851324
