L(s) = 1 | + 7.16·3-s − 5·5-s + 25.3·7-s + 24.3·9-s − 67.0·11-s − 2.41·13-s − 35.8·15-s − 12.5·17-s − 104.·19-s + 182.·21-s − 23·23-s + 25·25-s − 18.6·27-s + 221.·29-s + 102.·31-s − 480.·33-s − 126.·35-s + 2.56·37-s − 17.3·39-s − 89.4·41-s − 5.94·43-s − 121.·45-s − 549.·47-s + 301.·49-s − 89.6·51-s + 159.·53-s + 335.·55-s + ⋯ |
L(s) = 1 | + 1.37·3-s − 0.447·5-s + 1.37·7-s + 0.903·9-s − 1.83·11-s − 0.0515·13-s − 0.617·15-s − 0.178·17-s − 1.26·19-s + 1.89·21-s − 0.208·23-s + 0.200·25-s − 0.132·27-s + 1.41·29-s + 0.595·31-s − 2.53·33-s − 0.613·35-s + 0.0113·37-s − 0.0711·39-s − 0.340·41-s − 0.0210·43-s − 0.404·45-s − 1.70·47-s + 0.880·49-s − 0.246·51-s + 0.413·53-s + 0.821·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 + 23T \) |
good | 3 | \( 1 - 7.16T + 27T^{2} \) |
| 7 | \( 1 - 25.3T + 343T^{2} \) |
| 11 | \( 1 + 67.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 2.41T + 2.19e3T^{2} \) |
| 17 | \( 1 + 12.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 104.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 221.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 102.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 2.56T + 5.06e4T^{2} \) |
| 41 | \( 1 + 89.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 5.94T + 7.95e4T^{2} \) |
| 47 | \( 1 + 549.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 159.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 593.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 894.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 525.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 57.4T + 3.57e5T^{2} \) |
| 73 | \( 1 + 870.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 578.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 345.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 311.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.81e3T + 9.12e5T^{2} \) |
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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
−8.335075902226579391245801174146, −7.965391184126329822560219280871, −7.37951755964032502621915137405, −6.12066292610841025918411421433, −4.83669832610406974797280351887, −4.49855216910719728186847185516, −3.18532494781336736546536591987, −2.49332038576783861276190474851, −1.63753396738462646260982258939, 0,
1.63753396738462646260982258939, 2.49332038576783861276190474851, 3.18532494781336736546536591987, 4.49855216910719728186847185516, 4.83669832610406974797280351887, 6.12066292610841025918411421433, 7.37951755964032502621915137405, 7.965391184126329822560219280871, 8.335075902226579391245801174146