L(s) = 1 | − 2-s + 1.19·3-s + 4-s + 0.888·5-s − 1.19·6-s + 4.42·7-s − 8-s − 1.56·9-s − 0.888·10-s + 4.44·11-s + 1.19·12-s + 2.05·13-s − 4.42·14-s + 1.06·15-s + 16-s − 4.32·17-s + 1.56·18-s − 2.70·19-s + 0.888·20-s + 5.30·21-s − 4.44·22-s + 1.37·23-s − 1.19·24-s − 4.21·25-s − 2.05·26-s − 5.46·27-s + 4.42·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.691·3-s + 0.5·4-s + 0.397·5-s − 0.489·6-s + 1.67·7-s − 0.353·8-s − 0.521·9-s − 0.280·10-s + 1.34·11-s + 0.345·12-s + 0.569·13-s − 1.18·14-s + 0.274·15-s + 0.250·16-s − 1.04·17-s + 0.368·18-s − 0.620·19-s + 0.198·20-s + 1.15·21-s − 0.948·22-s + 0.286·23-s − 0.244·24-s − 0.842·25-s − 0.402·26-s − 1.05·27-s + 0.837·28-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
\(L(1)\) |
\(\approx\) |
\(2.487154806\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.487154806\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.19T + 3T^{2} \) |
| 5 | \( 1 - 0.888T + 5T^{2} \) |
| 7 | \( 1 - 4.42T + 7T^{2} \) |
| 11 | \( 1 - 4.44T + 11T^{2} \) |
| 13 | \( 1 - 2.05T + 13T^{2} \) |
| 17 | \( 1 + 4.32T + 17T^{2} \) |
| 19 | \( 1 + 2.70T + 19T^{2} \) |
| 23 | \( 1 - 1.37T + 23T^{2} \) |
| 29 | \( 1 - 6.67T + 29T^{2} \) |
| 31 | \( 1 + 1.09T + 31T^{2} \) |
| 37 | \( 1 - 4.69T + 37T^{2} \) |
| 41 | \( 1 - 4.58T + 41T^{2} \) |
| 43 | \( 1 - 3.98T + 43T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 + 6.24T + 53T^{2} \) |
| 59 | \( 1 - 7.04T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 - 0.0862T + 71T^{2} \) |
| 73 | \( 1 - 0.689T + 73T^{2} \) |
| 79 | \( 1 - 9.53T + 79T^{2} \) |
| 83 | \( 1 - 0.785T + 83T^{2} \) |
| 89 | \( 1 - 2.67T + 89T^{2} \) |
| 97 | \( 1 + 13.7T + 97T^{2} \) |
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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
−8.302273133314201646827887817770, −8.169438110034583629020053029577, −7.14567357956039225330938853854, −6.32416449224716619121909956932, −5.65119355740447550375063835775, −4.50487534307286094149852593358, −3.89290915759518737496045879332, −2.56925288973249227171824313961, −1.94016681822258102884528692138, −1.04332965934318713955715488403,
1.04332965934318713955715488403, 1.94016681822258102884528692138, 2.56925288973249227171824313961, 3.89290915759518737496045879332, 4.50487534307286094149852593358, 5.65119355740447550375063835775, 6.32416449224716619121909956932, 7.14567357956039225330938853854, 8.169438110034583629020053029577, 8.302273133314201646827887817770