| L(s) = 1 | − 2.18·2-s + 2.75·4-s − 4.62·7-s − 1.64·8-s + 2.53·11-s + 2.08·13-s + 10.0·14-s − 1.91·16-s − 0.481·17-s + 4.25·19-s − 5.52·22-s − 5.22·23-s − 4.55·26-s − 12.7·28-s + 29-s + 0.0174·31-s + 7.47·32-s + 1.05·34-s + 7.52·37-s − 9.28·38-s − 5.88·41-s − 4.91·43-s + 6.97·44-s + 11.3·46-s − 7.27·47-s + 14.3·49-s + 5.75·52-s + ⋯ |
| L(s) = 1 | − 1.54·2-s + 1.37·4-s − 1.74·7-s − 0.582·8-s + 0.763·11-s + 0.579·13-s + 2.69·14-s − 0.479·16-s − 0.116·17-s + 0.976·19-s − 1.17·22-s − 1.08·23-s − 0.893·26-s − 2.40·28-s + 0.185·29-s + 0.00313·31-s + 1.32·32-s + 0.180·34-s + 1.23·37-s − 1.50·38-s − 0.919·41-s − 0.749·43-s + 1.05·44-s + 1.67·46-s − 1.06·47-s + 2.05·49-s + 0.798·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 2.18T + 2T^{2} \) |
| 7 | \( 1 + 4.62T + 7T^{2} \) |
| 11 | \( 1 - 2.53T + 11T^{2} \) |
| 13 | \( 1 - 2.08T + 13T^{2} \) |
| 17 | \( 1 + 0.481T + 17T^{2} \) |
| 19 | \( 1 - 4.25T + 19T^{2} \) |
| 23 | \( 1 + 5.22T + 23T^{2} \) |
| 31 | \( 1 - 0.0174T + 31T^{2} \) |
| 37 | \( 1 - 7.52T + 37T^{2} \) |
| 41 | \( 1 + 5.88T + 41T^{2} \) |
| 43 | \( 1 + 4.91T + 43T^{2} \) |
| 47 | \( 1 + 7.27T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + 2.12T + 59T^{2} \) |
| 61 | \( 1 + 2.79T + 61T^{2} \) |
| 67 | \( 1 - 5.03T + 67T^{2} \) |
| 71 | \( 1 + 8.56T + 71T^{2} \) |
| 73 | \( 1 - 1.59T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 - 4.69T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 + 0.438T + 97T^{2} \) |
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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
−7.80196688871341799313587000744, −7.05021462862696119195030109830, −6.34771985663206283435171423749, −6.10929055805573924044056627915, −4.76646549018961584653092210045, −3.67736421132661765028341254671, −3.11773943872885147138048298139, −2.00242099554618918549460723305, −1.00044873542233122847911848721, 0,
1.00044873542233122847911848721, 2.00242099554618918549460723305, 3.11773943872885147138048298139, 3.67736421132661765028341254671, 4.76646549018961584653092210045, 6.10929055805573924044056627915, 6.34771985663206283435171423749, 7.05021462862696119195030109830, 7.80196688871341799313587000744
