L(s) = 1 | − 2-s + 3-s + 4-s − 0.561·5-s − 6-s + 7-s − 8-s + 9-s + 0.561·10-s + 1.56·11-s + 12-s − 14-s − 0.561·15-s + 16-s − 5·17-s − 18-s + 4.68·19-s − 0.561·20-s + 21-s − 1.56·22-s − 0.876·23-s − 24-s − 4.68·25-s + 27-s + 28-s − 29-s + 0.561·30-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.251·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.177·10-s + 0.470·11-s + 0.288·12-s − 0.267·14-s − 0.144·15-s + 0.250·16-s − 1.21·17-s − 0.235·18-s + 1.07·19-s − 0.125·20-s + 0.218·21-s − 0.332·22-s − 0.182·23-s − 0.204·24-s − 0.936·25-s + 0.192·27-s + 0.188·28-s − 0.185·29-s + 0.102·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 5 | \( 1 + 0.561T + 5T^{2} \) |
| 11 | \( 1 - 1.56T + 11T^{2} \) |
| 17 | \( 1 + 5T + 17T^{2} \) |
| 19 | \( 1 - 4.68T + 19T^{2} \) |
| 23 | \( 1 + 0.876T + 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 - 3.12T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 - 4.12T + 41T^{2} \) |
| 43 | \( 1 + 11.1T + 43T^{2} \) |
| 47 | \( 1 + 4.68T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 + 4.87T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 - 7.12T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 6.56T + 73T^{2} \) |
| 79 | \( 1 - 5.56T + 79T^{2} \) |
| 83 | \( 1 + 6.24T + 83T^{2} \) |
| 89 | \( 1 + 2.68T + 89T^{2} \) |
| 97 | \( 1 - 4.24T + 97T^{2} \) |
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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
−7.968683849399496586477314823813, −6.89465030640008403520547817601, −6.57645719526747876334572953452, −5.47385089066654520392255396046, −4.68471788222644084723319808800, −3.80195086230667835884483493133, −3.10097626876529295742675709419, −2.07155322849606364131641203530, −1.39902167106495990563657555931, 0,
1.39902167106495990563657555931, 2.07155322849606364131641203530, 3.10097626876529295742675709419, 3.80195086230667835884483493133, 4.68471788222644084723319808800, 5.47385089066654520392255396046, 6.57645719526747876334572953452, 6.89465030640008403520547817601, 7.968683849399496586477314823813