| L(s) = 1 | − 4·2-s − 3.22·3-s + 16·4-s + 12.8·6-s + 200.·7-s − 64·8-s − 232.·9-s − 530.·11-s − 51.5·12-s + 169·13-s − 803.·14-s + 256·16-s + 15.1·17-s + 930.·18-s − 392.·19-s − 647.·21-s + 2.12e3·22-s − 2.63e3·23-s + 206.·24-s − 676·26-s + 1.53e3·27-s + 3.21e3·28-s − 7.13e3·29-s + 6.82e3·31-s − 1.02e3·32-s + 1.71e3·33-s − 60.4·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.206·3-s + 0.5·4-s + 0.146·6-s + 1.54·7-s − 0.353·8-s − 0.957·9-s − 1.32·11-s − 0.103·12-s + 0.277·13-s − 1.09·14-s + 0.250·16-s + 0.0126·17-s + 0.676·18-s − 0.249·19-s − 0.320·21-s + 0.934·22-s − 1.03·23-s + 0.0731·24-s − 0.196·26-s + 0.404·27-s + 0.774·28-s − 1.57·29-s + 1.27·31-s − 0.176·32-s + 0.273·33-s − 0.00897·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.169461078\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.169461078\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - 169T \) |
| good | 3 | \( 1 + 3.22T + 243T^{2} \) |
| 7 | \( 1 - 200.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 530.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 15.1T + 1.41e6T^{2} \) |
| 19 | \( 1 + 392.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.63e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.13e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.82e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.32e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.21e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.10e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 9.25e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.52e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.03e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.42e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 7.07e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.62e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.10e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.94e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.75e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.33e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.20e3T + 8.58e9T^{2} \) |
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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
−9.827368131635018958990693248991, −8.700364875246430657269633530282, −8.030261061234225320760615063274, −7.58956219480603520471565611444, −6.08687649191492674689615270447, −5.40297087602086235323837428973, −4.35415902053394893686048004945, −2.76442694529660162709486327242, −1.87363729367961763016185673991, −0.56892626840803306579820719839,
0.56892626840803306579820719839, 1.87363729367961763016185673991, 2.76442694529660162709486327242, 4.35415902053394893686048004945, 5.40297087602086235323837428973, 6.08687649191492674689615270447, 7.58956219480603520471565611444, 8.030261061234225320760615063274, 8.700364875246430657269633530282, 9.827368131635018958990693248991
