| L(s) = 1 | − 37·7-s − 19·13-s − 163·19-s − 125·25-s + 308·31-s + 323·37-s − 520·43-s + 1.02e3·49-s + 719·61-s − 127·67-s − 919·73-s − 1.38e3·79-s + 703·91-s − 523·97-s − 1.80e3·103-s − 646·109-s + ⋯ |
| L(s) = 1 | − 1.99·7-s − 0.405·13-s − 1.96·19-s − 25-s + 1.78·31-s + 1.43·37-s − 1.84·43-s + 2.99·49-s + 1.50·61-s − 0.231·67-s − 1.47·73-s − 1.97·79-s + 0.809·91-s − 0.547·97-s − 1.72·103-s − 0.567·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + p^{3} T^{2} \) |
| 7 | \( 1 + 37 T + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 + 19 T + p^{3} T^{2} \) |
| 17 | \( 1 + p^{3} T^{2} \) |
| 19 | \( 1 + 163 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + p^{3} T^{2} \) |
| 31 | \( 1 - 308 T + p^{3} T^{2} \) |
| 37 | \( 1 - 323 T + p^{3} T^{2} \) |
| 41 | \( 1 + p^{3} T^{2} \) |
| 43 | \( 1 + 520 T + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 - 719 T + p^{3} T^{2} \) |
| 67 | \( 1 + 127 T + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 + 919 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1387 T + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 + p^{3} T^{2} \) |
| 97 | \( 1 + 523 T + p^{3} T^{2} \) |
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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
−12.86813110321959253387127591130, −11.81662014694048874974144999152, −10.29487780678826224561663286717, −9.659605363844428103309562395955, −8.396127786596266983197280986868, −6.81765522105100052143272051507, −6.05251640242681356935218892129, −4.14720765790314535805919714519, −2.70171297139772601612613454647, 0,
2.70171297139772601612613454647, 4.14720765790314535805919714519, 6.05251640242681356935218892129, 6.81765522105100052143272051507, 8.396127786596266983197280986868, 9.659605363844428103309562395955, 10.29487780678826224561663286717, 11.81662014694048874974144999152, 12.86813110321959253387127591130
