L(s) = 1 | − 2-s + 3-s + 4-s + 3.71·5-s − 6-s + 1.88·7-s − 8-s + 9-s − 3.71·10-s − 0.931·11-s + 12-s − 13-s − 1.88·14-s + 3.71·15-s + 16-s + 0.978·17-s − 18-s − 4.18·19-s + 3.71·20-s + 1.88·21-s + 0.931·22-s + 3.17·23-s − 24-s + 8.77·25-s + 26-s + 27-s + 1.88·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.65·5-s − 0.408·6-s + 0.711·7-s − 0.353·8-s + 0.333·9-s − 1.17·10-s − 0.280·11-s + 0.288·12-s − 0.277·13-s − 0.502·14-s + 0.958·15-s + 0.250·16-s + 0.237·17-s − 0.235·18-s − 0.960·19-s + 0.829·20-s + 0.410·21-s + 0.198·22-s + 0.661·23-s − 0.204·24-s + 1.75·25-s + 0.196·26-s + 0.192·27-s + 0.355·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.922188446\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.922188446\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
good | 5 | \( 1 - 3.71T + 5T^{2} \) |
| 7 | \( 1 - 1.88T + 7T^{2} \) |
| 11 | \( 1 + 0.931T + 11T^{2} \) |
| 17 | \( 1 - 0.978T + 17T^{2} \) |
| 19 | \( 1 + 4.18T + 19T^{2} \) |
| 23 | \( 1 - 3.17T + 23T^{2} \) |
| 29 | \( 1 + 7.70T + 29T^{2} \) |
| 31 | \( 1 - 0.646T + 31T^{2} \) |
| 37 | \( 1 - 0.528T + 37T^{2} \) |
| 41 | \( 1 - 9.05T + 41T^{2} \) |
| 43 | \( 1 - 9.80T + 43T^{2} \) |
| 47 | \( 1 - 5.18T + 47T^{2} \) |
| 53 | \( 1 - 1.64T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 + 8.72T + 61T^{2} \) |
| 67 | \( 1 - 6.40T + 67T^{2} \) |
| 71 | \( 1 + 4.44T + 71T^{2} \) |
| 73 | \( 1 - 5.43T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 + 0.0537T + 83T^{2} \) |
| 89 | \( 1 - 1.73T + 89T^{2} \) |
| 97 | \( 1 - 9.80T + 97T^{2} \) |
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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
−7.72206747988655815062274987098, −7.45735317092633446120164527064, −6.39459884666868567365862984963, −5.91282719838483499039264532781, −5.15114624850719758024687706254, −4.36861131052278534631855190593, −3.18703843018185410438789206432, −2.25369393321670025272606991022, −1.96728923559611254672493394692, −0.939532629374990433559932707760,
0.939532629374990433559932707760, 1.96728923559611254672493394692, 2.25369393321670025272606991022, 3.18703843018185410438789206432, 4.36861131052278534631855190593, 5.15114624850719758024687706254, 5.91282719838483499039264532781, 6.39459884666868567365862984963, 7.45735317092633446120164527064, 7.72206747988655815062274987098