L(s) = 1 | − 0.107·2-s + 3-s − 1.98·4-s − 4.11·5-s − 0.107·6-s − 2.41·7-s + 0.428·8-s + 9-s + 0.441·10-s + 3.22·11-s − 1.98·12-s + 1.21·13-s + 0.260·14-s − 4.11·15-s + 3.93·16-s − 17-s − 0.107·18-s − 3.26·19-s + 8.17·20-s − 2.41·21-s − 0.346·22-s − 0.375·23-s + 0.428·24-s + 11.8·25-s − 0.130·26-s + 27-s + 4.80·28-s + ⋯ |
L(s) = 1 | − 0.0760·2-s + 0.577·3-s − 0.994·4-s − 1.83·5-s − 0.0438·6-s − 0.914·7-s + 0.151·8-s + 0.333·9-s + 0.139·10-s + 0.971·11-s − 0.574·12-s + 0.336·13-s + 0.0694·14-s − 1.06·15-s + 0.982·16-s − 0.242·17-s − 0.0253·18-s − 0.748·19-s + 1.82·20-s − 0.527·21-s − 0.0738·22-s − 0.0783·23-s + 0.0875·24-s + 2.37·25-s − 0.0255·26-s + 0.192·27-s + 0.908·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 79 | \( 1 - T \) |
good | 2 | \( 1 + 0.107T + 2T^{2} \) |
| 5 | \( 1 + 4.11T + 5T^{2} \) |
| 7 | \( 1 + 2.41T + 7T^{2} \) |
| 11 | \( 1 - 3.22T + 11T^{2} \) |
| 13 | \( 1 - 1.21T + 13T^{2} \) |
| 19 | \( 1 + 3.26T + 19T^{2} \) |
| 23 | \( 1 + 0.375T + 23T^{2} \) |
| 29 | \( 1 + 2.01T + 29T^{2} \) |
| 31 | \( 1 - 6.34T + 31T^{2} \) |
| 37 | \( 1 - 8.38T + 37T^{2} \) |
| 41 | \( 1 + 5.02T + 41T^{2} \) |
| 43 | \( 1 - 8.14T + 43T^{2} \) |
| 47 | \( 1 - 0.107T + 47T^{2} \) |
| 53 | \( 1 - 13.2T + 53T^{2} \) |
| 59 | \( 1 + 5.52T + 59T^{2} \) |
| 61 | \( 1 + 9.40T + 61T^{2} \) |
| 67 | \( 1 + 13.1T + 67T^{2} \) |
| 71 | \( 1 - 8.17T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 83 | \( 1 + 1.92T + 83T^{2} \) |
| 89 | \( 1 + 9.78T + 89T^{2} \) |
| 97 | \( 1 - 7.20T + 97T^{2} \) |
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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
−8.138970113855934157619332373019, −7.56637126642976114528275354078, −6.75797707695031027630718106424, −5.97930195424929831353276983035, −4.56373409553872453062497672195, −4.19041218727973862554614901307, −3.60101252413551080357993676227, −2.84685567204689012062249470074, −1.07189289662706409509893950424, 0,
1.07189289662706409509893950424, 2.84685567204689012062249470074, 3.60101252413551080357993676227, 4.19041218727973862554614901307, 4.56373409553872453062497672195, 5.97930195424929831353276983035, 6.75797707695031027630718106424, 7.56637126642976114528275354078, 8.138970113855934157619332373019