| L(s) = 1 | − 17·7-s − 89·13-s + 107·19-s − 125·25-s − 308·31-s + 433·37-s − 520·43-s − 54·49-s + 901·61-s + 1.00e3·67-s − 271·73-s − 503·79-s + 1.51e3·91-s + 1.85e3·97-s + 19·103-s + 646·109-s + ⋯ |
| L(s) = 1 | − 0.917·7-s − 1.89·13-s + 1.29·19-s − 25-s − 1.78·31-s + 1.92·37-s − 1.84·43-s − 0.157·49-s + 1.89·61-s + 1.83·67-s − 0.434·73-s − 0.716·79-s + 1.74·91-s + 1.93·97-s + 0.0181·103-s + 0.567·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.108121998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.108121998\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 17 T + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 + 89 T + p^{3} T^{2} \) |
| 17 | \( 1 + p^{3} T^{2} \) |
| 19 | \( 1 - 107 T + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + p^{3} T^{2} \) |
| 31 | \( 1 + 308 T + p^{3} T^{2} \) |
| 37 | \( 1 - 433 T + p^{3} T^{2} \) |
| 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 | \( 1 - 901 T + p^{3} T^{2} \) |
| 67 | \( 1 - 1007 T + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 + 271 T + p^{3} T^{2} \) |
| 79 | \( 1 + 503 T + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 + p^{3} T^{2} \) |
| 97 | \( 1 - 1853 T + p^{3} T^{2} \) |
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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
−9.248982234201922318792715246870, −8.003434440796912499995913756313, −7.35916865329540936144755477195, −6.68563884751336951455133263371, −5.63447289157440386424856232762, −4.97814753607766147666682150362, −3.83495244125911976832144702162, −2.96144556766361583451666124319, −2.01294365286918779549346745125, −0.47546435069009155152390318785,
0.47546435069009155152390318785, 2.01294365286918779549346745125, 2.96144556766361583451666124319, 3.83495244125911976832144702162, 4.97814753607766147666682150362, 5.63447289157440386424856232762, 6.68563884751336951455133263371, 7.35916865329540936144755477195, 8.003434440796912499995913756313, 9.248982234201922318792715246870
