L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 4·7-s − 8-s + 9-s + 12-s + 13-s + 4·14-s + 16-s − 18-s + 5·19-s − 4·21-s − 24-s − 26-s + 27-s − 4·28-s + 3·29-s − 4·31-s − 32-s + 36-s − 7·37-s − 5·38-s + 39-s + 3·41-s + 4·42-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.277·13-s + 1.06·14-s + 1/4·16-s − 0.235·18-s + 1.14·19-s − 0.872·21-s − 0.204·24-s − 0.196·26-s + 0.192·27-s − 0.755·28-s + 0.557·29-s − 0.718·31-s − 0.176·32-s + 1/6·36-s − 1.15·37-s − 0.811·38-s + 0.160·39-s + 0.468·41-s + 0.617·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.261653142\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.261653142\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−9.078339016738156498544771857711, −8.696657713782855691873839819406, −7.55763254790499154685263093337, −7.04556559344328990168385895474, −6.21942354210226036235913961670, −5.34655100340507626856083927354, −3.86541777052856340745686880083, −3.21110717019141316546603900405, −2.28203067662838759760760090270, −0.805107178230786108851722746580,
0.805107178230786108851722746580, 2.28203067662838759760760090270, 3.21110717019141316546603900405, 3.86541777052856340745686880083, 5.34655100340507626856083927354, 6.21942354210226036235913961670, 7.04556559344328990168385895474, 7.55763254790499154685263093337, 8.696657713782855691873839819406, 9.078339016738156498544771857711