L(s) = 1 | + 0.241·2-s − 0.709·3-s − 0.941·4-s + 5-s − 0.170·6-s + 1.77·7-s − 0.468·8-s − 0.497·9-s + 0.241·10-s + 1.13·11-s + 0.667·12-s + 13-s + 0.426·14-s − 0.709·15-s + 0.829·16-s − 1.94·17-s − 0.119·18-s − 0.941·20-s − 1.25·21-s + 0.273·22-s − 1.49·23-s + 0.332·24-s + 25-s + 0.241·26-s + 1.06·27-s − 1.66·28-s − 0.170·30-s + ⋯ |
L(s) = 1 | + 0.241·2-s − 0.709·3-s − 0.941·4-s + 5-s − 0.170·6-s + 1.77·7-s − 0.468·8-s − 0.497·9-s + 0.241·10-s + 1.13·11-s + 0.667·12-s + 13-s + 0.426·14-s − 0.709·15-s + 0.829·16-s − 1.94·17-s − 0.119·18-s − 0.941·20-s − 1.25·21-s + 0.273·22-s − 1.49·23-s + 0.332·24-s + 25-s + 0.241·26-s + 1.06·27-s − 1.66·28-s − 0.170·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.191515107\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.191515107\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 - T \) |
good | 2 | \( 1 - 0.241T + T^{2} \) |
| 3 | \( 1 + 0.709T + T^{2} \) |
| 7 | \( 1 - 1.77T + T^{2} \) |
| 11 | \( 1 - 1.13T + T^{2} \) |
| 17 | \( 1 + 1.94T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.49T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 0.241T + T^{2} \) |
| 47 | \( 1 + 1.49T + T^{2} \) |
| 53 | \( 1 - 1.13T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.13T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 1.77T + T^{2} \) |
| 97 | \( 1 + 0.709T + T^{2} \) |
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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
−9.126548559153460701727290078381, −8.642964435789660811574300282855, −8.106720715796559263682245744456, −6.55727357299554117322076225485, −6.11369646462744035558499401746, −5.25436372414274366527085012873, −4.61913681788934639303492886440, −3.91264759869244484481188417881, −2.24675019914236999261926051050, −1.21301305312917594204842932823,
1.21301305312917594204842932823, 2.24675019914236999261926051050, 3.91264759869244484481188417881, 4.61913681788934639303492886440, 5.25436372414274366527085012873, 6.11369646462744035558499401746, 6.55727357299554117322076225485, 8.106720715796559263682245744456, 8.642964435789660811574300282855, 9.126548559153460701727290078381