L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 8-s + 9-s − 10-s + 4·11-s + 12-s − 2·13-s + 15-s + 16-s + 2·17-s − 18-s + 20-s − 4·22-s + 23-s − 24-s + 25-s + 2·26-s + 27-s + 6·29-s − 30-s − 32-s + 4·33-s − 2·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s + 0.288·12-s − 0.554·13-s + 0.258·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 0.223·20-s − 0.852·22-s + 0.208·23-s − 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 1.11·29-s − 0.182·30-s − 0.176·32-s + 0.696·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.541395905\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.541395905\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−10.09863317860401952638074389853, −9.630715428080546025209075211314, −8.807813628420990463208948046699, −8.052767886224194001355832864579, −7.00827699878192292867604115541, −6.32649531451210752196473227385, −5.02067976449257282596483927763, −3.70894573273403509375184849934, −2.52811735526220925449002095844, −1.27803220748287546016129683063,
1.27803220748287546016129683063, 2.52811735526220925449002095844, 3.70894573273403509375184849934, 5.02067976449257282596483927763, 6.32649531451210752196473227385, 7.00827699878192292867604115541, 8.052767886224194001355832864579, 8.807813628420990463208948046699, 9.630715428080546025209075211314, 10.09863317860401952638074389853