| L(s) = 1 | + 3-s − 0.476·7-s + 9-s − 3.34·11-s − 13-s − 17-s − 6.77·19-s − 0.476·21-s + 4.77·23-s + 27-s + 7.59·29-s + 7.93·31-s − 3.34·33-s − 7.82·37-s − 39-s + 9.46·41-s − 1.82·43-s + 9.06·47-s − 6.77·49-s − 51-s − 9.50·53-s − 6.77·57-s + 3.16·59-s − 9.59·61-s − 0.476·63-s + 1.34·67-s + 4.77·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.180·7-s + 0.333·9-s − 1.00·11-s − 0.277·13-s − 0.242·17-s − 1.55·19-s − 0.103·21-s + 0.995·23-s + 0.192·27-s + 1.41·29-s + 1.42·31-s − 0.582·33-s − 1.28·37-s − 0.160·39-s + 1.47·41-s − 0.277·43-s + 1.32·47-s − 0.967·49-s − 0.140·51-s − 1.30·53-s − 0.897·57-s + 0.411·59-s − 1.22·61-s − 0.0600·63-s + 0.164·67-s + 0.574·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 0.476T + 7T^{2} \) |
| 11 | \( 1 + 3.34T + 11T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 19 | \( 1 + 6.77T + 19T^{2} \) |
| 23 | \( 1 - 4.77T + 23T^{2} \) |
| 29 | \( 1 - 7.59T + 29T^{2} \) |
| 31 | \( 1 - 7.93T + 31T^{2} \) |
| 37 | \( 1 + 7.82T + 37T^{2} \) |
| 41 | \( 1 - 9.46T + 41T^{2} \) |
| 43 | \( 1 + 1.82T + 43T^{2} \) |
| 47 | \( 1 - 9.06T + 47T^{2} \) |
| 53 | \( 1 + 9.50T + 53T^{2} \) |
| 59 | \( 1 - 3.16T + 59T^{2} \) |
| 61 | \( 1 + 9.59T + 61T^{2} \) |
| 67 | \( 1 - 1.34T + 67T^{2} \) |
| 71 | \( 1 + 4.86T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 + 2.17T + 79T^{2} \) |
| 83 | \( 1 + 9.16T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 + 7.04T + 97T^{2} \) |
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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
−7.64836977191603915429201226211, −6.74826729799244326616851944656, −6.34684616556637364500724730651, −5.27231316095327368916784673723, −4.64394502821784250997703741371, −3.96455204586088306593940314204, −2.75294442567611409536773755567, −2.61398239626680768403055755465, −1.33759323868721965817113338273, 0,
1.33759323868721965817113338273, 2.61398239626680768403055755465, 2.75294442567611409536773755567, 3.96455204586088306593940314204, 4.64394502821784250997703741371, 5.27231316095327368916784673723, 6.34684616556637364500724730651, 6.74826729799244326616851944656, 7.64836977191603915429201226211
