| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 2.82·7-s − 8-s + 9-s − 5.65·11-s − 12-s + 13-s + 2.82·14-s + 16-s + 4.82·17-s − 18-s − 2.82·19-s + 2.82·21-s + 5.65·22-s − 8.48·23-s + 24-s − 26-s − 27-s − 2.82·28-s − 3.17·29-s + 4·31-s − 32-s + 5.65·33-s − 4.82·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.408·6-s − 1.06·7-s − 0.353·8-s + 0.333·9-s − 1.70·11-s − 0.288·12-s + 0.277·13-s + 0.755·14-s + 0.250·16-s + 1.17·17-s − 0.235·18-s − 0.648·19-s + 0.617·21-s + 1.20·22-s − 1.76·23-s + 0.204·24-s − 0.196·26-s − 0.192·27-s − 0.534·28-s − 0.588·29-s + 0.718·31-s − 0.176·32-s + 0.984·33-s − 0.828·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5296676554\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5296676554\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 + 5.65T + 11T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 8.48T + 23T^{2} \) |
| 29 | \( 1 + 3.17T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 0.343T + 37T^{2} \) |
| 41 | \( 1 - 3.65T + 41T^{2} \) |
| 43 | \( 1 - 1.65T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 9.31T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 + 2.48T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 + 17.6T + 83T^{2} \) |
| 89 | \( 1 + 4.34T + 89T^{2} \) |
| 97 | \( 1 - 8.82T + 97T^{2} \) |
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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
−9.381119483580321670414397939315, −8.243409452173305058300395212591, −7.75916640616896204146584076146, −6.88064381881974669853439768497, −5.92468674655933683902335830682, −5.56731529714752289199474351991, −4.21175510665677260094557632268, −3.13046651485550034973767298423, −2.15400126718019764932313057866, −0.52886766222493632297846208002,
0.52886766222493632297846208002, 2.15400126718019764932313057866, 3.13046651485550034973767298423, 4.21175510665677260094557632268, 5.56731529714752289199474351991, 5.92468674655933683902335830682, 6.88064381881974669853439768497, 7.75916640616896204146584076146, 8.243409452173305058300395212591, 9.381119483580321670414397939315
