L(s) = 1 | + 2-s − 3-s + 4-s − 0.585·5-s − 6-s + 8-s + 9-s − 0.585·10-s − 0.414·11-s − 12-s + 13-s + 0.585·15-s + 16-s − 2.41·17-s + 18-s − 2.17·19-s − 0.585·20-s − 0.414·22-s + 1.41·23-s − 24-s − 4.65·25-s + 26-s − 27-s − 1.82·29-s + 0.585·30-s − 8.48·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.261·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.185·10-s − 0.124·11-s − 0.288·12-s + 0.277·13-s + 0.151·15-s + 0.250·16-s − 0.585·17-s + 0.235·18-s − 0.498·19-s − 0.130·20-s − 0.0883·22-s + 0.294·23-s − 0.204·24-s − 0.931·25-s + 0.196·26-s − 0.192·27-s − 0.339·29-s + 0.106·30-s − 1.52·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3822 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 + 0.585T + 5T^{2} \) |
| 11 | \( 1 + 0.414T + 11T^{2} \) |
| 17 | \( 1 + 2.41T + 17T^{2} \) |
| 19 | \( 1 + 2.17T + 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 + 1.82T + 29T^{2} \) |
| 31 | \( 1 + 8.48T + 31T^{2} \) |
| 37 | \( 1 - 1.41T + 37T^{2} \) |
| 41 | \( 1 + 9.89T + 41T^{2} \) |
| 43 | \( 1 - 6.48T + 43T^{2} \) |
| 47 | \( 1 - T + 47T^{2} \) |
| 53 | \( 1 - 9.48T + 53T^{2} \) |
| 59 | \( 1 - 2.07T + 59T^{2} \) |
| 61 | \( 1 + 4.41T + 61T^{2} \) |
| 67 | \( 1 - 1.82T + 67T^{2} \) |
| 71 | \( 1 + 5T + 71T^{2} \) |
| 73 | \( 1 - 1.41T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 - 7.65T + 83T^{2} \) |
| 89 | \( 1 - 2.58T + 89T^{2} \) |
| 97 | \( 1 + 0.928T + 97T^{2} \) |
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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
−7.962251865651287606204258679649, −7.19108399046494015444868927495, −6.58905287116292104554837693630, −5.73954196444726087468153789032, −5.21847147520853028522622065069, −4.21713916522509687324786626422, −3.73359280838118878535217068790, −2.55118735194606936406090659544, −1.55303991959244434776358613732, 0,
1.55303991959244434776358613732, 2.55118735194606936406090659544, 3.73359280838118878535217068790, 4.21713916522509687324786626422, 5.21847147520853028522622065069, 5.73954196444726087468153789032, 6.58905287116292104554837693630, 7.19108399046494015444868927495, 7.962251865651287606204258679649