L(s) = 1 | + 2-s − 3-s + 4-s − 4.41·5-s − 6-s + 8-s + 9-s − 4.41·10-s − 11-s − 12-s − 2.82·13-s + 4.41·15-s + 16-s + 4.24·17-s + 18-s + 19-s − 4.41·20-s − 22-s + 0.585·23-s − 24-s + 14.4·25-s − 2.82·26-s − 27-s + 7.82·29-s + 4.41·30-s + 0.656·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.97·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 1.39·10-s − 0.301·11-s − 0.288·12-s − 0.784·13-s + 1.13·15-s + 0.250·16-s + 1.02·17-s + 0.235·18-s + 0.229·19-s − 0.987·20-s − 0.213·22-s + 0.122·23-s − 0.204·24-s + 2.89·25-s − 0.554·26-s − 0.192·27-s + 1.45·29-s + 0.805·30-s + 0.117·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + 4.41T + 5T^{2} \) |
| 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 - 4.24T + 17T^{2} \) |
| 23 | \( 1 - 0.585T + 23T^{2} \) |
| 29 | \( 1 - 7.82T + 29T^{2} \) |
| 31 | \( 1 - 0.656T + 31T^{2} \) |
| 37 | \( 1 + 8.24T + 37T^{2} \) |
| 41 | \( 1 + 3.41T + 41T^{2} \) |
| 43 | \( 1 + 2.24T + 43T^{2} \) |
| 47 | \( 1 - 8.82T + 47T^{2} \) |
| 53 | \( 1 + 3.82T + 53T^{2} \) |
| 59 | \( 1 + 8.07T + 59T^{2} \) |
| 61 | \( 1 - 0.828T + 61T^{2} \) |
| 67 | \( 1 + 2T + 67T^{2} \) |
| 71 | \( 1 - 11.8T + 71T^{2} \) |
| 73 | \( 1 + 9.17T + 73T^{2} \) |
| 79 | \( 1 + 5.82T + 79T^{2} \) |
| 83 | \( 1 - 11.1T + 83T^{2} \) |
| 89 | \( 1 - 12.2T + 89T^{2} \) |
| 97 | \( 1 + 8.41T + 97T^{2} \) |
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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
−7.69455050728402810596011300212, −7.07234083213947470372541896923, −6.47227086400524367552556999968, −5.29816877415669655321696242430, −4.89562429515684703647924779871, −4.14077065403950660365487494320, −3.42342812643037481036938482380, −2.73363027048482320977641619538, −1.14098873665669935040871863935, 0,
1.14098873665669935040871863935, 2.73363027048482320977641619538, 3.42342812643037481036938482380, 4.14077065403950660365487494320, 4.89562429515684703647924779871, 5.29816877415669655321696242430, 6.47227086400524367552556999968, 7.07234083213947470372541896923, 7.69455050728402810596011300212