L(s) = 1 | + 1.36·2-s + 3-s − 0.134·4-s + 0.714·5-s + 1.36·6-s − 2.88·7-s − 2.91·8-s + 9-s + 0.975·10-s − 0.0240·11-s − 0.134·12-s + 2.58·13-s − 3.94·14-s + 0.714·15-s − 3.71·16-s − 17-s + 1.36·18-s + 3.62·19-s − 0.0963·20-s − 2.88·21-s − 0.0328·22-s − 2.49·23-s − 2.91·24-s − 4.48·25-s + 3.52·26-s + 27-s + 0.388·28-s + ⋯ |
L(s) = 1 | + 0.965·2-s + 0.577·3-s − 0.0673·4-s + 0.319·5-s + 0.557·6-s − 1.09·7-s − 1.03·8-s + 0.333·9-s + 0.308·10-s − 0.00725·11-s − 0.0389·12-s + 0.716·13-s − 1.05·14-s + 0.184·15-s − 0.928·16-s − 0.242·17-s + 0.321·18-s + 0.831·19-s − 0.0215·20-s − 0.629·21-s − 0.00700·22-s − 0.521·23-s − 0.595·24-s − 0.897·25-s + 0.691·26-s + 0.192·27-s + 0.0734·28-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 | \( 1 - 1.36T + 2T^{2} \) |
| 5 | \( 1 - 0.714T + 5T^{2} \) |
| 7 | \( 1 + 2.88T + 7T^{2} \) |
| 11 | \( 1 + 0.0240T + 11T^{2} \) |
| 13 | \( 1 - 2.58T + 13T^{2} \) |
| 19 | \( 1 - 3.62T + 19T^{2} \) |
| 23 | \( 1 + 2.49T + 23T^{2} \) |
| 29 | \( 1 + 3.33T + 29T^{2} \) |
| 31 | \( 1 + 2.93T + 31T^{2} \) |
| 37 | \( 1 + 10.1T + 37T^{2} \) |
| 41 | \( 1 - 0.780T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 9.28T + 47T^{2} \) |
| 53 | \( 1 + 1.97T + 53T^{2} \) |
| 59 | \( 1 + 14.3T + 59T^{2} \) |
| 61 | \( 1 + 3.55T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 - 6.82T + 71T^{2} \) |
| 73 | \( 1 - 6.11T + 73T^{2} \) |
| 83 | \( 1 + 5.96T + 83T^{2} \) |
| 89 | \( 1 + 9.36T + 89T^{2} \) |
| 97 | \( 1 - 8.44T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.072960715866595262886476147560, −7.23901055008334912742305359850, −6.31732407767780933215478878843, −5.88687544390601727943631614625, −5.05414762241241156218418801168, −4.05600976051669295472737041064, −3.48681751367345332329354876023, −2.87612476593940192131027695622, −1.70575008863852479173060944198, 0,
1.70575008863852479173060944198, 2.87612476593940192131027695622, 3.48681751367345332329354876023, 4.05600976051669295472737041064, 5.05414762241241156218418801168, 5.88687544390601727943631614625, 6.31732407767780933215478878843, 7.23901055008334912742305359850, 8.072960715866595262886476147560