L(s) = 1 | − 1.61·2-s + 0.618·4-s + 7-s + 2.23·8-s − 4.23·11-s − 3.23·13-s − 1.61·14-s − 4.85·16-s + 6.47·17-s + 4.47·19-s + 6.85·22-s − 1.76·23-s + 5.23·26-s + 0.618·28-s − 5·29-s − 9.70·31-s + 3.38·32-s − 10.4·34-s + 3·37-s − 7.23·38-s − 9.23·41-s + 6.23·43-s − 2.61·44-s + 2.85·46-s + 2·47-s + 49-s − 2.00·52-s + ⋯ |
L(s) = 1 | − 1.14·2-s + 0.309·4-s + 0.377·7-s + 0.790·8-s − 1.27·11-s − 0.897·13-s − 0.432·14-s − 1.21·16-s + 1.56·17-s + 1.02·19-s + 1.46·22-s − 0.367·23-s + 1.02·26-s + 0.116·28-s − 0.928·29-s − 1.74·31-s + 0.597·32-s − 1.79·34-s + 0.493·37-s − 1.17·38-s − 1.44·41-s + 0.950·43-s − 0.394·44-s + 0.420·46-s + 0.291·47-s + 0.142·49-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 11 | \( 1 + 4.23T + 11T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 - 6.47T + 17T^{2} \) |
| 19 | \( 1 - 4.47T + 19T^{2} \) |
| 23 | \( 1 + 1.76T + 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + 9.70T + 31T^{2} \) |
| 37 | \( 1 - 3T + 37T^{2} \) |
| 41 | \( 1 + 9.23T + 41T^{2} \) |
| 43 | \( 1 - 6.23T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 - 0.472T + 53T^{2} \) |
| 59 | \( 1 - 1.70T + 59T^{2} \) |
| 61 | \( 1 - 3.70T + 61T^{2} \) |
| 67 | \( 1 - 0.236T + 67T^{2} \) |
| 71 | \( 1 - 4.70T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 + 5.70T + 83T^{2} \) |
| 89 | \( 1 + 12.7T + 89T^{2} \) |
| 97 | \( 1 - 0.763T + 97T^{2} \) |
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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
−9.152324080860552896873138965367, −8.091469851647682618347906093718, −7.66039773680149360912047345536, −7.14652234007855425244534984426, −5.49108109204687166760062541317, −5.17365756280936866046728723114, −3.84309314686664368207717091926, −2.59568940076681454060396923237, −1.42478666260907303296009538598, 0,
1.42478666260907303296009538598, 2.59568940076681454060396923237, 3.84309314686664368207717091926, 5.17365756280936866046728723114, 5.49108109204687166760062541317, 7.14652234007855425244534984426, 7.66039773680149360912047345536, 8.091469851647682618347906093718, 9.152324080860552896873138965367