| L(s) = 1 | + 1.27·2-s + 3-s − 0.384·4-s + 1.27·6-s + 0.483·7-s − 3.03·8-s + 9-s − 0.384·12-s + 3.14·13-s + 0.614·14-s − 3.08·16-s − 2.51·17-s + 1.27·18-s − 3.72·19-s + 0.483·21-s + 8.53·23-s − 3.03·24-s + 3.99·26-s + 27-s − 0.185·28-s − 9.98·29-s − 5.78·31-s + 2.14·32-s − 3.19·34-s − 0.384·36-s + 3.20·37-s − 4.72·38-s + ⋯ |
| L(s) = 1 | + 0.898·2-s + 0.577·3-s − 0.192·4-s + 0.518·6-s + 0.182·7-s − 1.07·8-s + 0.333·9-s − 0.111·12-s + 0.872·13-s + 0.164·14-s − 0.770·16-s − 0.608·17-s + 0.299·18-s − 0.853·19-s + 0.105·21-s + 1.77·23-s − 0.618·24-s + 0.783·26-s + 0.192·27-s − 0.0351·28-s − 1.85·29-s − 1.03·31-s + 0.379·32-s − 0.547·34-s − 0.0641·36-s + 0.527·37-s − 0.766·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.27T + 2T^{2} \) |
| 7 | \( 1 - 0.483T + 7T^{2} \) |
| 13 | \( 1 - 3.14T + 13T^{2} \) |
| 17 | \( 1 + 2.51T + 17T^{2} \) |
| 19 | \( 1 + 3.72T + 19T^{2} \) |
| 23 | \( 1 - 8.53T + 23T^{2} \) |
| 29 | \( 1 + 9.98T + 29T^{2} \) |
| 31 | \( 1 + 5.78T + 31T^{2} \) |
| 37 | \( 1 - 3.20T + 37T^{2} \) |
| 41 | \( 1 + 7.86T + 41T^{2} \) |
| 43 | \( 1 + 5.97T + 43T^{2} \) |
| 47 | \( 1 + 6.13T + 47T^{2} \) |
| 53 | \( 1 - 1.58T + 53T^{2} \) |
| 59 | \( 1 + 0.377T + 59T^{2} \) |
| 61 | \( 1 - 3.00T + 61T^{2} \) |
| 67 | \( 1 + 5.98T + 67T^{2} \) |
| 71 | \( 1 - 8.34T + 71T^{2} \) |
| 73 | \( 1 + 0.151T + 73T^{2} \) |
| 79 | \( 1 - 7.48T + 79T^{2} \) |
| 83 | \( 1 - 4.05T + 83T^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 + 3.60T + 97T^{2} \) |
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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
−7.23980611290080458811607553393, −6.63935671821335519928071640349, −5.92447081646321119274640749614, −5.11782782460578241711876085014, −4.63226247551807518749196370454, −3.66921238182464638194816182878, −3.42704174991659492421519494812, −2.38012605046993913819898222166, −1.47371998128030219370969057322, 0,
1.47371998128030219370969057322, 2.38012605046993913819898222166, 3.42704174991659492421519494812, 3.66921238182464638194816182878, 4.63226247551807518749196370454, 5.11782782460578241711876085014, 5.92447081646321119274640749614, 6.63935671821335519928071640349, 7.23980611290080458811607553393
