| L(s) = 1 | − 5·2-s + 17·4-s − 45·8-s − 27·9-s − 68·11-s + 89·16-s + 135·18-s + 340·22-s − 40·23-s − 125·25-s − 166·29-s − 85·32-s − 459·36-s + 450·37-s − 180·43-s − 1.15e3·44-s + 200·46-s + 625·50-s + 590·53-s + 830·58-s − 287·64-s − 740·67-s + 688·71-s + 1.21e3·72-s − 2.25e3·74-s − 1.38e3·79-s + 729·81-s + ⋯ |
| L(s) = 1 | − 1.76·2-s + 17/8·4-s − 1.98·8-s − 9-s − 1.86·11-s + 1.39·16-s + 1.76·18-s + 3.29·22-s − 0.362·23-s − 25-s − 1.06·29-s − 0.469·32-s − 2.12·36-s + 1.99·37-s − 0.638·43-s − 3.96·44-s + 0.641·46-s + 1.76·50-s + 1.52·53-s + 1.87·58-s − 0.560·64-s − 1.34·67-s + 1.15·71-s + 1.98·72-s − 3.53·74-s − 1.97·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + 5 T + p^{3} T^{2} \) |
| 3 | \( 1 + p^{3} T^{2} \) |
| 5 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 + 68 T + p^{3} T^{2} \) |
| 13 | \( 1 + p^{3} T^{2} \) |
| 17 | \( 1 + p^{3} T^{2} \) |
| 19 | \( 1 + p^{3} T^{2} \) |
| 23 | \( 1 + 40 T + p^{3} T^{2} \) |
| 29 | \( 1 + 166 T + p^{3} T^{2} \) |
| 31 | \( 1 + p^{3} T^{2} \) |
| 37 | \( 1 - 450 T + p^{3} T^{2} \) |
| 41 | \( 1 + p^{3} T^{2} \) |
| 43 | \( 1 + 180 T + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 - 590 T + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 + p^{3} T^{2} \) |
| 67 | \( 1 + 740 T + p^{3} T^{2} \) |
| 71 | \( 1 - 688 T + p^{3} T^{2} \) |
| 73 | \( 1 + p^{3} T^{2} \) |
| 79 | \( 1 + 1384 T + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 + p^{3} T^{2} \) |
| 97 | \( 1 + p^{3} T^{2} \) |
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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
−15.02894184714965737715717038841, −13.30442063818121698660187280692, −11.64156989511880964719127413578, −10.69837535176370904724193150519, −9.657416614835928522773169291177, −8.353921336728576853261425161019, −7.56866168292611020750437653852, −5.77862461637823030407687672667, −2.50258280312093899836671041842, 0,
2.50258280312093899836671041842, 5.77862461637823030407687672667, 7.56866168292611020750437653852, 8.353921336728576853261425161019, 9.657416614835928522773169291177, 10.69837535176370904724193150519, 11.64156989511880964719127413578, 13.30442063818121698660187280692, 15.02894184714965737715717038841
