L(s) = 1 | + 1.10·2-s − 3-s − 0.768·4-s − 5-s − 1.10·6-s − 3.69·7-s − 3.07·8-s + 9-s − 1.10·10-s + 2.03·11-s + 0.768·12-s − 1.40·13-s − 4.09·14-s + 15-s − 1.87·16-s + 4.49·17-s + 1.10·18-s + 0.460·19-s + 0.768·20-s + 3.69·21-s + 2.26·22-s + 3.56·23-s + 3.07·24-s + 25-s − 1.55·26-s − 27-s + 2.83·28-s + ⋯ |
L(s) = 1 | + 0.784·2-s − 0.577·3-s − 0.384·4-s − 0.447·5-s − 0.453·6-s − 1.39·7-s − 1.08·8-s + 0.333·9-s − 0.350·10-s + 0.614·11-s + 0.221·12-s − 0.388·13-s − 1.09·14-s + 0.258·15-s − 0.468·16-s + 1.08·17-s + 0.261·18-s + 0.105·19-s + 0.171·20-s + 0.805·21-s + 0.482·22-s + 0.744·23-s + 0.627·24-s + 0.200·25-s − 0.305·26-s − 0.192·27-s + 0.536·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
good | 2 | \( 1 - 1.10T + 2T^{2} \) |
| 7 | \( 1 + 3.69T + 7T^{2} \) |
| 11 | \( 1 - 2.03T + 11T^{2} \) |
| 13 | \( 1 + 1.40T + 13T^{2} \) |
| 17 | \( 1 - 4.49T + 17T^{2} \) |
| 19 | \( 1 - 0.460T + 19T^{2} \) |
| 23 | \( 1 - 3.56T + 23T^{2} \) |
| 29 | \( 1 - 4.74T + 29T^{2} \) |
| 31 | \( 1 + 0.540T + 31T^{2} \) |
| 37 | \( 1 - 1.42T + 37T^{2} \) |
| 41 | \( 1 + 6.91T + 41T^{2} \) |
| 43 | \( 1 - 5.00T + 43T^{2} \) |
| 47 | \( 1 + 7.18T + 47T^{2} \) |
| 53 | \( 1 - 7.50T + 53T^{2} \) |
| 59 | \( 1 - 6.34T + 59T^{2} \) |
| 61 | \( 1 + 8.61T + 61T^{2} \) |
| 67 | \( 1 + 0.0657T + 67T^{2} \) |
| 71 | \( 1 - 6.54T + 71T^{2} \) |
| 73 | \( 1 + 11.7T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 + 1.01T + 83T^{2} \) |
| 89 | \( 1 - 8.31T + 89T^{2} \) |
| 97 | \( 1 + 3.51T + 97T^{2} \) |
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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
−7.50066832745316776073691115368, −6.77107708971075264808049213226, −6.23567441890598554328194663641, −5.51986619697580220332425407209, −4.84452222902162435153925232954, −4.04105866151721454775493730833, −3.38904738341354925895069788706, −2.79096371194785029644658834601, −1.05163653646860264801906349188, 0,
1.05163653646860264801906349188, 2.79096371194785029644658834601, 3.38904738341354925895069788706, 4.04105866151721454775493730833, 4.84452222902162435153925232954, 5.51986619697580220332425407209, 6.23567441890598554328194663641, 6.77107708971075264808049213226, 7.50066832745316776073691115368