| L(s) = 1 | + 1.87·2-s − 3-s + 1.53·4-s + 2.15·5-s − 1.87·6-s − 7-s − 0.882·8-s + 9-s + 4.05·10-s − 1.05·11-s − 1.53·12-s − 1.95·13-s − 1.87·14-s − 2.15·15-s − 4.71·16-s − 3.66·17-s + 1.87·18-s − 5.80·19-s + 3.30·20-s + 21-s − 1.97·22-s + 0.881·23-s + 0.882·24-s − 0.340·25-s − 3.67·26-s − 27-s − 1.53·28-s + ⋯ |
| L(s) = 1 | + 1.32·2-s − 0.577·3-s + 0.765·4-s + 0.965·5-s − 0.767·6-s − 0.377·7-s − 0.312·8-s + 0.333·9-s + 1.28·10-s − 0.317·11-s − 0.441·12-s − 0.542·13-s − 0.502·14-s − 0.557·15-s − 1.17·16-s − 0.888·17-s + 0.442·18-s − 1.33·19-s + 0.738·20-s + 0.218·21-s − 0.421·22-s + 0.183·23-s + 0.180·24-s − 0.0681·25-s − 0.720·26-s − 0.192·27-s − 0.289·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 1.87T + 2T^{2} \) |
| 5 | \( 1 - 2.15T + 5T^{2} \) |
| 11 | \( 1 + 1.05T + 11T^{2} \) |
| 13 | \( 1 + 1.95T + 13T^{2} \) |
| 17 | \( 1 + 3.66T + 17T^{2} \) |
| 19 | \( 1 + 5.80T + 19T^{2} \) |
| 23 | \( 1 - 0.881T + 23T^{2} \) |
| 29 | \( 1 + 7.03T + 29T^{2} \) |
| 31 | \( 1 - 9.43T + 31T^{2} \) |
| 37 | \( 1 + 3.58T + 37T^{2} \) |
| 41 | \( 1 + 8.10T + 41T^{2} \) |
| 43 | \( 1 + 1.22T + 43T^{2} \) |
| 47 | \( 1 - 4.52T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 4.86T + 59T^{2} \) |
| 61 | \( 1 + 9.78T + 61T^{2} \) |
| 67 | \( 1 + 13.0T + 67T^{2} \) |
| 71 | \( 1 + 3.03T + 71T^{2} \) |
| 73 | \( 1 + 0.269T + 73T^{2} \) |
| 79 | \( 1 - 5.50T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 - 17.5T + 97T^{2} \) |
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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
−8.615887243160281031083091153933, −7.30740248255502489961694417975, −6.49094862759982490661194638944, −6.07769777039727105457337356922, −5.30488827261585092901531447882, −4.64719208695928108833138725765, −3.86763740759772128603199485397, −2.70020002687447218662870811891, −1.96589836437539483247588572116, 0,
1.96589836437539483247588572116, 2.70020002687447218662870811891, 3.86763740759772128603199485397, 4.64719208695928108833138725765, 5.30488827261585092901531447882, 6.07769777039727105457337356922, 6.49094862759982490661194638944, 7.30740248255502489961694417975, 8.615887243160281031083091153933
