L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 1.04·7-s − 8-s + 9-s − 10-s − 1.35·11-s + 12-s − 1.04·14-s + 15-s + 16-s − 1.08·17-s − 18-s − 2.93·19-s + 20-s + 1.04·21-s + 1.35·22-s − 0.692·23-s − 24-s + 25-s + 27-s + 1.04·28-s − 2.37·29-s − 30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.447·5-s − 0.408·6-s + 0.396·7-s − 0.353·8-s + 0.333·9-s − 0.316·10-s − 0.409·11-s + 0.288·12-s − 0.280·14-s + 0.258·15-s + 0.250·16-s − 0.263·17-s − 0.235·18-s − 0.674·19-s + 0.223·20-s + 0.228·21-s + 0.289·22-s − 0.144·23-s − 0.204·24-s + 0.200·25-s + 0.192·27-s + 0.198·28-s − 0.441·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 1.04T + 7T^{2} \) |
| 11 | \( 1 + 1.35T + 11T^{2} \) |
| 17 | \( 1 + 1.08T + 17T^{2} \) |
| 19 | \( 1 + 2.93T + 19T^{2} \) |
| 23 | \( 1 + 0.692T + 23T^{2} \) |
| 29 | \( 1 + 2.37T + 29T^{2} \) |
| 31 | \( 1 + 9.85T + 31T^{2} \) |
| 37 | \( 1 + 9.26T + 37T^{2} \) |
| 41 | \( 1 + 2.84T + 41T^{2} \) |
| 43 | \( 1 + 4.45T + 43T^{2} \) |
| 47 | \( 1 - 3.31T + 47T^{2} \) |
| 53 | \( 1 - 0.664T + 53T^{2} \) |
| 59 | \( 1 - 1.96T + 59T^{2} \) |
| 61 | \( 1 - 3.24T + 61T^{2} \) |
| 67 | \( 1 + 6.91T + 67T^{2} \) |
| 71 | \( 1 + 2.29T + 71T^{2} \) |
| 73 | \( 1 + 3.36T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 2.68T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 + 1.56T + 97T^{2} \) |
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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
−8.030388823059868280120344212138, −7.24320572168585258275483639072, −6.69849021219321335991623294129, −5.71483097439053974511835318230, −5.04636533501717605646594529686, −3.99288714035540905078431788199, −3.13006931986189083828914130790, −2.12292900676721814489047872300, −1.59158949651019088302826124550, 0,
1.59158949651019088302826124550, 2.12292900676721814489047872300, 3.13006931986189083828914130790, 3.99288714035540905078431788199, 5.04636533501717605646594529686, 5.71483097439053974511835318230, 6.69849021219321335991623294129, 7.24320572168585258275483639072, 8.030388823059868280120344212138