| L(s) = 1 | − 4.53·2-s + 6.46·3-s + 12.5·4-s − 29.3·6-s − 20.7·8-s + 14.8·9-s − 54.0·11-s + 81.2·12-s + 75.2·13-s − 6.60·16-s − 71.2·17-s − 67.1·18-s + 65.5·19-s + 245.·22-s − 125.·23-s − 133.·24-s − 341.·26-s − 78.8·27-s + 190.·29-s + 193.·31-s + 195.·32-s − 349.·33-s + 323.·34-s + 186.·36-s − 114.·37-s − 297.·38-s + 486.·39-s + ⋯ |
| L(s) = 1 | − 1.60·2-s + 1.24·3-s + 1.57·4-s − 1.99·6-s − 0.915·8-s + 0.548·9-s − 1.48·11-s + 1.95·12-s + 1.60·13-s − 0.103·16-s − 1.01·17-s − 0.879·18-s + 0.791·19-s + 2.37·22-s − 1.13·23-s − 1.13·24-s − 2.57·26-s − 0.561·27-s + 1.21·29-s + 1.11·31-s + 1.08·32-s − 1.84·33-s + 1.62·34-s + 0.861·36-s − 0.509·37-s − 1.26·38-s + 1.99·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 4.53T + 8T^{2} \) |
| 3 | \( 1 - 6.46T + 27T^{2} \) |
| 11 | \( 1 + 54.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 75.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 71.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 65.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 125.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 190.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 193.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 114.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 216.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 413.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 113.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 584.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 203.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 162.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 477.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 822.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 798.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 468.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 310.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.31e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.31e3T + 9.12e5T^{2} \) |
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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
−8.761707611861864407467289205371, −8.229742407989804852429502504390, −7.896818233633476095318847107744, −6.86549271268285509763267693644, −5.88900146734510807040959223837, −4.43289551386949779488541864361, −3.13985392336095749798502308397, −2.38923472622279067456399828313, −1.35088420252162443431386266723, 0,
1.35088420252162443431386266723, 2.38923472622279067456399828313, 3.13985392336095749798502308397, 4.43289551386949779488541864361, 5.88900146734510807040959223837, 6.86549271268285509763267693644, 7.896818233633476095318847107744, 8.229742407989804852429502504390, 8.761707611861864407467289205371
