| L(s) = 1 | − 3-s + 5-s − 4.78·7-s + 9-s + 1.53·11-s − 4.69·13-s − 15-s + 2.92·17-s + 6.94·19-s + 4.78·21-s + 2.12·23-s + 25-s − 27-s − 4.88·29-s + 2.53·31-s − 1.53·33-s − 4.78·35-s − 1.09·37-s + 4.69·39-s − 6.75·41-s + 11.1·43-s + 45-s + 0.205·47-s + 15.8·49-s − 2.92·51-s − 8.49·53-s + 1.53·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.80·7-s + 0.333·9-s + 0.462·11-s − 1.30·13-s − 0.258·15-s + 0.710·17-s + 1.59·19-s + 1.04·21-s + 0.442·23-s + 0.200·25-s − 0.192·27-s − 0.906·29-s + 0.454·31-s − 0.266·33-s − 0.808·35-s − 0.180·37-s + 0.751·39-s − 1.05·41-s + 1.69·43-s + 0.149·45-s + 0.0299·47-s + 2.27·49-s − 0.410·51-s − 1.16·53-s + 0.206·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| good | 7 | \( 1 + 4.78T + 7T^{2} \) |
| 11 | \( 1 - 1.53T + 11T^{2} \) |
| 13 | \( 1 + 4.69T + 13T^{2} \) |
| 17 | \( 1 - 2.92T + 17T^{2} \) |
| 19 | \( 1 - 6.94T + 19T^{2} \) |
| 23 | \( 1 - 2.12T + 23T^{2} \) |
| 29 | \( 1 + 4.88T + 29T^{2} \) |
| 31 | \( 1 - 2.53T + 31T^{2} \) |
| 37 | \( 1 + 1.09T + 37T^{2} \) |
| 41 | \( 1 + 6.75T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 0.205T + 47T^{2} \) |
| 53 | \( 1 + 8.49T + 53T^{2} \) |
| 59 | \( 1 + 1.17T + 59T^{2} \) |
| 61 | \( 1 - 5.62T + 61T^{2} \) |
| 71 | \( 1 - 6.31T + 71T^{2} \) |
| 73 | \( 1 + 2.16T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 14.7T + 83T^{2} \) |
| 89 | \( 1 - 1.02T + 89T^{2} \) |
| 97 | \( 1 + 15.1T + 97T^{2} \) |
| 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
−7.906231624180832538634231508784, −7.01574813177216796383098565128, −6.78376108639056913458008118445, −5.67997706034459002622138781115, −5.43384395731943298222452690168, −4.25669042746221372007407980732, −3.28584366723292151347069881303, −2.67264093593231255249595307296, −1.22746292667416931846844606418, 0,
1.22746292667416931846844606418, 2.67264093593231255249595307296, 3.28584366723292151347069881303, 4.25669042746221372007407980732, 5.43384395731943298222452690168, 5.67997706034459002622138781115, 6.78376108639056913458008118445, 7.01574813177216796383098565128, 7.906231624180832538634231508784
