| L(s) = 1 | + 2.17·2-s − 3·3-s − 3.26·4-s − 5·5-s − 6.52·6-s − 20.6·7-s − 24.5·8-s + 9·9-s − 10.8·10-s + 9.80·12-s + 74.6·13-s − 44.8·14-s + 15·15-s − 27.1·16-s − 44.4·17-s + 19.5·18-s + 9.00·19-s + 16.3·20-s + 61.9·21-s + 125.·23-s + 73.5·24-s + 25·25-s + 162.·26-s − 27·27-s + 67.4·28-s − 156.·29-s + 32.6·30-s + ⋯ |
| L(s) = 1 | + 0.769·2-s − 0.577·3-s − 0.408·4-s − 0.447·5-s − 0.443·6-s − 1.11·7-s − 1.08·8-s + 0.333·9-s − 0.343·10-s + 0.235·12-s + 1.59·13-s − 0.856·14-s + 0.258·15-s − 0.424·16-s − 0.634·17-s + 0.256·18-s + 0.108·19-s + 0.182·20-s + 0.643·21-s + 1.13·23-s + 0.625·24-s + 0.200·25-s + 1.22·26-s − 0.192·27-s + 0.455·28-s − 1.00·29-s + 0.198·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 3T \) |
| 5 | \( 1 + 5T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.17T + 8T^{2} \) |
| 7 | \( 1 + 20.6T + 343T^{2} \) |
| 13 | \( 1 - 74.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 44.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 9.00T + 6.85e3T^{2} \) |
| 23 | \( 1 - 125.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 156.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 221.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 80.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 158.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 95.8T + 7.95e4T^{2} \) |
| 47 | \( 1 - 569.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 9.77T + 1.48e5T^{2} \) |
| 59 | \( 1 + 192.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 48.3T + 2.26e5T^{2} \) |
| 67 | \( 1 + 561.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 426.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 142.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 693.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 836.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 86.9T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.16e3T + 9.12e5T^{2} \) |
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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
−8.804123193314957536044544875396, −7.57228778452239187764450962898, −6.52453029718272754618293612163, −6.12269076063800576084471021209, −5.24557287850880657051770531416, −4.28404260606597207493828879429, −3.65212095439596895617725446507, −2.84185573977877501571762939646, −1.02571384537801128734670489997, 0,
1.02571384537801128734670489997, 2.84185573977877501571762939646, 3.65212095439596895617725446507, 4.28404260606597207493828879429, 5.24557287850880657051770531416, 6.12269076063800576084471021209, 6.52453029718272754618293612163, 7.57228778452239187764450962898, 8.804123193314957536044544875396
