| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 5·5-s + 6·6-s − 18·7-s − 8·8-s + 9·9-s − 10·10-s − 70·11-s − 12·12-s − 86·13-s + 36·14-s − 15·15-s + 16·16-s − 56·17-s − 18·18-s + 108·19-s + 20·20-s + 54·21-s + 140·22-s + 23·23-s + 24·24-s + 25·25-s + 172·26-s − 27·27-s − 72·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.971·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.91·11-s − 0.288·12-s − 1.83·13-s + 0.687·14-s − 0.258·15-s + 1/4·16-s − 0.798·17-s − 0.235·18-s + 1.30·19-s + 0.223·20-s + 0.561·21-s + 1.35·22-s + 0.208·23-s + 0.204·24-s + 1/5·25-s + 1.29·26-s − 0.192·27-s − 0.485·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4212883201\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4212883201\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 - p T \) |
| 23 | \( 1 - p T \) |
| good | 7 | \( 1 + 18 T + p^{3} T^{2} \) |
| 11 | \( 1 + 70 T + p^{3} T^{2} \) |
| 13 | \( 1 + 86 T + p^{3} T^{2} \) |
| 17 | \( 1 + 56 T + p^{3} T^{2} \) |
| 19 | \( 1 - 108 T + p^{3} T^{2} \) |
| 29 | \( 1 - 186 T + p^{3} T^{2} \) |
| 31 | \( 1 + 120 T + p^{3} T^{2} \) |
| 37 | \( 1 + 232 T + p^{3} T^{2} \) |
| 41 | \( 1 + 398 T + p^{3} T^{2} \) |
| 43 | \( 1 - 120 T + p^{3} T^{2} \) |
| 47 | \( 1 - 88 T + p^{3} T^{2} \) |
| 53 | \( 1 + 190 T + p^{3} T^{2} \) |
| 59 | \( 1 - 696 T + p^{3} T^{2} \) |
| 61 | \( 1 - 504 T + p^{3} T^{2} \) |
| 67 | \( 1 - 432 T + p^{3} T^{2} \) |
| 71 | \( 1 + 72 T + p^{3} T^{2} \) |
| 73 | \( 1 - 102 T + p^{3} T^{2} \) |
| 79 | \( 1 + 218 T + p^{3} T^{2} \) |
| 83 | \( 1 - 82 T + p^{3} T^{2} \) |
| 89 | \( 1 + 828 T + p^{3} T^{2} \) |
| 97 | \( 1 - 650 T + p^{3} T^{2} \) |
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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
−10.02620379878255090235769772218, −9.549204356786542087200742786997, −8.348080483335426454609700419571, −7.28736774020315572882360965276, −6.80048748710540171503270753362, −5.49421060876912485844279398001, −4.96732529273475142061966442975, −3.06637500294113358373399097042, −2.23293426252862947769738087125, −0.39864138147190799985066623172,
0.39864138147190799985066623172, 2.23293426252862947769738087125, 3.06637500294113358373399097042, 4.96732529273475142061966442975, 5.49421060876912485844279398001, 6.80048748710540171503270753362, 7.28736774020315572882360965276, 8.348080483335426454609700419571, 9.549204356786542087200742786997, 10.02620379878255090235769772218
