| L(s) = 1 | − 2-s + 3-s + 4-s − 1.56·5-s − 6-s − 3.98·7-s − 8-s + 9-s + 1.56·10-s + 2.46·11-s + 12-s + 6.11·13-s + 3.98·14-s − 1.56·15-s + 16-s − 3.60·17-s − 18-s − 4.79·19-s − 1.56·20-s − 3.98·21-s − 2.46·22-s + 23-s − 24-s − 2.56·25-s − 6.11·26-s + 27-s − 3.98·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.698·5-s − 0.408·6-s − 1.50·7-s − 0.353·8-s + 0.333·9-s + 0.493·10-s + 0.743·11-s + 0.288·12-s + 1.69·13-s + 1.06·14-s − 0.403·15-s + 0.250·16-s − 0.874·17-s − 0.235·18-s − 1.09·19-s − 0.349·20-s − 0.869·21-s − 0.525·22-s + 0.208·23-s − 0.204·24-s − 0.512·25-s − 1.19·26-s + 0.192·27-s − 0.752·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.082191341\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.082191341\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 + 1.56T + 5T^{2} \) |
| 7 | \( 1 + 3.98T + 7T^{2} \) |
| 11 | \( 1 - 2.46T + 11T^{2} \) |
| 13 | \( 1 - 6.11T + 13T^{2} \) |
| 17 | \( 1 + 3.60T + 17T^{2} \) |
| 19 | \( 1 + 4.79T + 19T^{2} \) |
| 31 | \( 1 + 3.90T + 31T^{2} \) |
| 37 | \( 1 - 4.61T + 37T^{2} \) |
| 41 | \( 1 - 3.01T + 41T^{2} \) |
| 43 | \( 1 + 2.59T + 43T^{2} \) |
| 47 | \( 1 - 8.41T + 47T^{2} \) |
| 53 | \( 1 + 2.45T + 53T^{2} \) |
| 59 | \( 1 - 9.18T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 - 6.45T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 - 1.24T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 - 8.96T + 83T^{2} \) |
| 89 | \( 1 - 3.54T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−8.575461244041919230859130616399, −7.895156362183642195990070781579, −6.95717309091734308813574443851, −6.45203622355342278772406495555, −5.89255186242387812094568331112, −4.21056237963743952730465038818, −3.78071530237853460264202644882, −3.00662467895420760662769086181, −1.92386203276090569725609376749, −0.63298066853845904301300507442,
0.63298066853845904301300507442, 1.92386203276090569725609376749, 3.00662467895420760662769086181, 3.78071530237853460264202644882, 4.21056237963743952730465038818, 5.89255186242387812094568331112, 6.45203622355342278772406495555, 6.95717309091734308813574443851, 7.895156362183642195990070781579, 8.575461244041919230859130616399
