| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 5·7-s + 8-s + 9-s − 10-s + 11-s + 12-s − 4·13-s + 5·14-s − 15-s + 16-s − 17-s + 18-s − 4·19-s − 20-s + 5·21-s + 22-s − 3·23-s + 24-s + 25-s − 4·26-s + 27-s + 5·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.88·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s − 1.10·13-s + 1.33·14-s − 0.258·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.917·19-s − 0.223·20-s + 1.09·21-s + 0.213·22-s − 0.625·23-s + 0.204·24-s + 1/5·25-s − 0.784·26-s + 0.192·27-s + 0.944·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.648979502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.648979502\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−8.108525295885192359622752373158, −7.51110708678346099833819859599, −6.82252553212373593199766565072, −5.89437595238359938013372722115, −4.90264149496210911108332991445, −4.49106307738197976760337451924, −3.97180352327347755988112978926, −2.64782541743752884095701152637, −2.17743716244594118029465053129, −1.06589430873563926322594809177,
1.06589430873563926322594809177, 2.17743716244594118029465053129, 2.64782541743752884095701152637, 3.97180352327347755988112978926, 4.49106307738197976760337451924, 4.90264149496210911108332991445, 5.89437595238359938013372722115, 6.82252553212373593199766565072, 7.51110708678346099833819859599, 8.108525295885192359622752373158
