| L(s) = 1 | − 2.50·2-s − 0.817·3-s + 4.29·4-s + 2.05·6-s + 7-s − 5.75·8-s − 2.33·9-s + 3.50·11-s − 3.50·12-s + 0.541·13-s − 2.50·14-s + 5.84·16-s − 0.856·17-s + 5.84·18-s − 3.98·19-s − 0.817·21-s − 8.80·22-s + 23-s + 4.70·24-s − 1.35·26-s + 4.35·27-s + 4.29·28-s + 0.916·29-s + 0.314·31-s − 3.14·32-s − 2.86·33-s + 2.14·34-s + ⋯ |
| L(s) = 1 | − 1.77·2-s − 0.471·3-s + 2.14·4-s + 0.837·6-s + 0.377·7-s − 2.03·8-s − 0.777·9-s + 1.05·11-s − 1.01·12-s + 0.150·13-s − 0.670·14-s + 1.46·16-s − 0.207·17-s + 1.37·18-s − 0.913·19-s − 0.178·21-s − 1.87·22-s + 0.208·23-s + 0.959·24-s − 0.266·26-s + 0.838·27-s + 0.811·28-s + 0.170·29-s + 0.0565·31-s − 0.556·32-s − 0.499·33-s + 0.368·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 2.50T + 2T^{2} \) |
| 3 | \( 1 + 0.817T + 3T^{2} \) |
| 11 | \( 1 - 3.50T + 11T^{2} \) |
| 13 | \( 1 - 0.541T + 13T^{2} \) |
| 17 | \( 1 + 0.856T + 17T^{2} \) |
| 19 | \( 1 + 3.98T + 19T^{2} \) |
| 29 | \( 1 - 0.916T + 29T^{2} \) |
| 31 | \( 1 - 0.314T + 31T^{2} \) |
| 37 | \( 1 - 3.44T + 37T^{2} \) |
| 41 | \( 1 + 3.60T + 41T^{2} \) |
| 43 | \( 1 + 12.6T + 43T^{2} \) |
| 47 | \( 1 - 9.91T + 47T^{2} \) |
| 53 | \( 1 + 3.29T + 53T^{2} \) |
| 59 | \( 1 + 6.76T + 59T^{2} \) |
| 61 | \( 1 + 3.19T + 61T^{2} \) |
| 67 | \( 1 + 0.788T + 67T^{2} \) |
| 71 | \( 1 + 3.25T + 71T^{2} \) |
| 73 | \( 1 - 5.04T + 73T^{2} \) |
| 79 | \( 1 - 6.73T + 79T^{2} \) |
| 83 | \( 1 - 3.37T + 83T^{2} \) |
| 89 | \( 1 + 4.40T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−8.306375823726063605672189387181, −7.56647810330440734523925712907, −6.55187643857582543654180334985, −6.39939931787606755698226973461, −5.31327085850430989727043274995, −4.25245487365455191128967705450, −3.04988772516664253131280767091, −2.03457575611669316733552170143, −1.13811787906921475872862509639, 0,
1.13811787906921475872862509639, 2.03457575611669316733552170143, 3.04988772516664253131280767091, 4.25245487365455191128967705450, 5.31327085850430989727043274995, 6.39939931787606755698226973461, 6.55187643857582543654180334985, 7.56647810330440734523925712907, 8.306375823726063605672189387181
