| L(s) = 1 | − 2.78·2-s − 0.216·4-s − 5·5-s + 22.9·8-s + 13.9·10-s − 0.731·11-s + 63.9·13-s − 62.2·16-s + 29.6·17-s + 43.7·19-s + 1.08·20-s + 2.03·22-s − 157.·23-s + 25·25-s − 178.·26-s − 192.·29-s − 24.2·31-s − 9.79·32-s − 82.6·34-s − 62.1·37-s − 122.·38-s − 114.·40-s + 65.8·41-s + 97.8·43-s + 0.158·44-s + 440.·46-s − 432.·47-s + ⋯ |
| L(s) = 1 | − 0.986·2-s − 0.0270·4-s − 0.447·5-s + 1.01·8-s + 0.441·10-s − 0.0200·11-s + 1.36·13-s − 0.972·16-s + 0.422·17-s + 0.528·19-s + 0.0121·20-s + 0.0197·22-s − 1.43·23-s + 0.200·25-s − 1.34·26-s − 1.23·29-s − 0.140·31-s − 0.0541·32-s − 0.416·34-s − 0.276·37-s − 0.521·38-s − 0.453·40-s + 0.250·41-s + 0.346·43-s + 0.000542·44-s + 1.41·46-s − 1.34·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 2.78T + 8T^{2} \) |
| 11 | \( 1 + 0.731T + 1.33e3T^{2} \) |
| 13 | \( 1 - 63.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 29.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 43.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 157.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 192.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 24.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 62.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 65.8T + 6.89e4T^{2} \) |
| 43 | \( 1 - 97.8T + 7.95e4T^{2} \) |
| 47 | \( 1 + 432.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 301.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 350.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 405.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 853.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 214.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.13e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 51.4T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.41e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 266.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 546.T + 9.12e5T^{2} \) |
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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
−8.230073383323773394334745690437, −7.893959985564239831925317610214, −7.01276565127463858324240389151, −6.04339026130201765659091656559, −5.16401247786354522477401896922, −4.06010505026465708088774875408, −3.48570354239871170089203425499, −1.95659557402730080321952053600, −1.03667806148878539676228267671, 0,
1.03667806148878539676228267671, 1.95659557402730080321952053600, 3.48570354239871170089203425499, 4.06010505026465708088774875408, 5.16401247786354522477401896922, 6.04339026130201765659091656559, 7.01276565127463858324240389151, 7.893959985564239831925317610214, 8.230073383323773394334745690437
