| L(s) = 1 | − 2.26·2-s − 3-s + 3.11·4-s + 0.694·5-s + 2.26·6-s − 1.74·7-s − 2.51·8-s + 9-s − 1.57·10-s − 4.36·11-s − 3.11·12-s − 3.28·13-s + 3.95·14-s − 0.694·15-s − 0.539·16-s − 5.78·17-s − 2.26·18-s − 7.37·19-s + 2.16·20-s + 1.74·21-s + 9.87·22-s + 1.73·23-s + 2.51·24-s − 4.51·25-s + 7.43·26-s − 27-s − 5.43·28-s + ⋯ |
| L(s) = 1 | − 1.59·2-s − 0.577·3-s + 1.55·4-s + 0.310·5-s + 0.923·6-s − 0.660·7-s − 0.888·8-s + 0.333·9-s − 0.496·10-s − 1.31·11-s − 0.898·12-s − 0.912·13-s + 1.05·14-s − 0.179·15-s − 0.134·16-s − 1.40·17-s − 0.532·18-s − 1.69·19-s + 0.483·20-s + 0.381·21-s + 2.10·22-s + 0.361·23-s + 0.513·24-s − 0.903·25-s + 1.45·26-s − 0.192·27-s − 1.02·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2523 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1393936225\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1393936225\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 2.26T + 2T^{2} \) |
| 5 | \( 1 - 0.694T + 5T^{2} \) |
| 7 | \( 1 + 1.74T + 7T^{2} \) |
| 11 | \( 1 + 4.36T + 11T^{2} \) |
| 13 | \( 1 + 3.28T + 13T^{2} \) |
| 17 | \( 1 + 5.78T + 17T^{2} \) |
| 19 | \( 1 + 7.37T + 19T^{2} \) |
| 23 | \( 1 - 1.73T + 23T^{2} \) |
| 31 | \( 1 + 5.92T + 31T^{2} \) |
| 37 | \( 1 + 2.89T + 37T^{2} \) |
| 41 | \( 1 - 7.12T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 9.52T + 47T^{2} \) |
| 53 | \( 1 - 12.3T + 53T^{2} \) |
| 59 | \( 1 + 6.64T + 59T^{2} \) |
| 61 | \( 1 - 1.89T + 61T^{2} \) |
| 67 | \( 1 + 4.62T + 67T^{2} \) |
| 71 | \( 1 - 4.65T + 71T^{2} \) |
| 73 | \( 1 - 0.735T + 73T^{2} \) |
| 79 | \( 1 - 6.69T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 - 1.66T + 89T^{2} \) |
| 97 | \( 1 - 5.67T + 97T^{2} \) |
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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
−9.133601124478686159484656208220, −8.187813752208292704297183858249, −7.51095504332248203052347023500, −6.77527772136984894286583937119, −6.14767334532275086717226719086, −5.13464491826478077062494674994, −4.17693834732586700787194652462, −2.54383706118574103476761035182, −2.01767181238809234884165062594, −0.28655037356573287564753533370,
0.28655037356573287564753533370, 2.01767181238809234884165062594, 2.54383706118574103476761035182, 4.17693834732586700787194652462, 5.13464491826478077062494674994, 6.14767334532275086717226719086, 6.77527772136984894286583937119, 7.51095504332248203052347023500, 8.187813752208292704297183858249, 9.133601124478686159484656208220
