L(s) = 1 | + 0.273·2-s − 2.65·3-s − 1.92·4-s − 0.726·6-s + 0.273·7-s − 1.07·8-s + 4.02·9-s + 1.37·11-s + 5.10·12-s + 2.75·13-s + 0.0750·14-s + 3.55·16-s − 2·17-s + 1.10·18-s + 4.20·19-s − 0.726·21-s + 0.377·22-s − 8.05·23-s + 2.84·24-s + 0.754·26-s − 2.72·27-s − 0.527·28-s − 4.20·29-s + 10.8·31-s + 3.12·32-s − 3.65·33-s − 0.547·34-s + ⋯ |
L(s) = 1 | + 0.193·2-s − 1.53·3-s − 0.962·4-s − 0.296·6-s + 0.103·7-s − 0.380·8-s + 1.34·9-s + 0.415·11-s + 1.47·12-s + 0.763·13-s + 0.0200·14-s + 0.888·16-s − 0.485·17-s + 0.260·18-s + 0.965·19-s − 0.158·21-s + 0.0804·22-s − 1.67·23-s + 0.581·24-s + 0.147·26-s − 0.524·27-s − 0.0996·28-s − 0.781·29-s + 1.95·31-s + 0.552·32-s − 0.635·33-s − 0.0939·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 43 | \( 1 - T \) |
good | 2 | \( 1 - 0.273T + 2T^{2} \) |
| 3 | \( 1 + 2.65T + 3T^{2} \) |
| 7 | \( 1 - 0.273T + 7T^{2} \) |
| 11 | \( 1 - 1.37T + 11T^{2} \) |
| 13 | \( 1 - 2.75T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 - 4.20T + 19T^{2} \) |
| 23 | \( 1 + 8.05T + 23T^{2} \) |
| 29 | \( 1 + 4.20T + 29T^{2} \) |
| 31 | \( 1 - 10.8T + 31T^{2} \) |
| 37 | \( 1 + 5.92T + 37T^{2} \) |
| 41 | \( 1 + 6.70T + 41T^{2} \) |
| 47 | \( 1 + 7.45T + 47T^{2} \) |
| 53 | \( 1 + 1.45T + 53T^{2} \) |
| 59 | \( 1 + 0.0467T + 59T^{2} \) |
| 61 | \( 1 + 6.75T + 61T^{2} \) |
| 67 | \( 1 + 9.84T + 67T^{2} \) |
| 71 | \( 1 - 5.30T + 71T^{2} \) |
| 73 | \( 1 + 6.29T + 73T^{2} \) |
| 79 | \( 1 + 0.281T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 - 0.341T + 89T^{2} \) |
| 97 | \( 1 - 10.0T + 97T^{2} \) |
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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
−9.701005043650513197421202159067, −8.673645484408755591266140779729, −7.83580662927383766026305075298, −6.54321931882518448441836522775, −6.01036178110323389928621591950, −5.13744577293414473109578810289, −4.44092097252771101015376611628, −3.46880369277035457865994342617, −1.37483901160064691420287455163, 0,
1.37483901160064691420287455163, 3.46880369277035457865994342617, 4.44092097252771101015376611628, 5.13744577293414473109578810289, 6.01036178110323389928621591950, 6.54321931882518448441836522775, 7.83580662927383766026305075298, 8.673645484408755591266140779729, 9.701005043650513197421202159067