L(s) = 1 | − 2-s + 0.327·3-s + 4-s − 1.35·5-s − 0.327·6-s + 3.43·7-s − 8-s − 2.89·9-s + 1.35·10-s + 5.09·11-s + 0.327·12-s − 3.01·13-s − 3.43·14-s − 0.443·15-s + 16-s + 7.56·17-s + 2.89·18-s − 6.52·19-s − 1.35·20-s + 1.12·21-s − 5.09·22-s − 5.33·23-s − 0.327·24-s − 3.15·25-s + 3.01·26-s − 1.92·27-s + 3.43·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.188·3-s + 0.5·4-s − 0.606·5-s − 0.133·6-s + 1.29·7-s − 0.353·8-s − 0.964·9-s + 0.429·10-s + 1.53·11-s + 0.0944·12-s − 0.835·13-s − 0.918·14-s − 0.114·15-s + 0.250·16-s + 1.83·17-s + 0.681·18-s − 1.49·19-s − 0.303·20-s + 0.245·21-s − 1.08·22-s − 1.11·23-s − 0.0667·24-s − 0.631·25-s + 0.591·26-s − 0.370·27-s + 0.649·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2003 | \( 1 + T \) |
good | 3 | \( 1 - 0.327T + 3T^{2} \) |
| 5 | \( 1 + 1.35T + 5T^{2} \) |
| 7 | \( 1 - 3.43T + 7T^{2} \) |
| 11 | \( 1 - 5.09T + 11T^{2} \) |
| 13 | \( 1 + 3.01T + 13T^{2} \) |
| 17 | \( 1 - 7.56T + 17T^{2} \) |
| 19 | \( 1 + 6.52T + 19T^{2} \) |
| 23 | \( 1 + 5.33T + 23T^{2} \) |
| 29 | \( 1 + 4.11T + 29T^{2} \) |
| 31 | \( 1 - 3.27T + 31T^{2} \) |
| 37 | \( 1 + 9.67T + 37T^{2} \) |
| 41 | \( 1 + 8.83T + 41T^{2} \) |
| 43 | \( 1 + 1.26T + 43T^{2} \) |
| 47 | \( 1 - 6.16T + 47T^{2} \) |
| 53 | \( 1 + 6.20T + 53T^{2} \) |
| 59 | \( 1 + 0.844T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + 3.77T + 67T^{2} \) |
| 71 | \( 1 + 0.624T + 71T^{2} \) |
| 73 | \( 1 - 7.05T + 73T^{2} \) |
| 79 | \( 1 - 5.34T + 79T^{2} \) |
| 83 | \( 1 + 7.15T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 13.2T + 97T^{2} \) |
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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
−8.297244567323708931958900150213, −7.60624318221155310076304435670, −6.81365985203159686651824018091, −5.90671317398183638552003360770, −5.15917227416355103435409934789, −4.09686827954779649646750735521, −3.46624489982157746910540270969, −2.18578890383263228109598431935, −1.44532468184315241351087286240, 0,
1.44532468184315241351087286240, 2.18578890383263228109598431935, 3.46624489982157746910540270969, 4.09686827954779649646750735521, 5.15917227416355103435409934789, 5.90671317398183638552003360770, 6.81365985203159686651824018091, 7.60624318221155310076304435670, 8.297244567323708931958900150213