| L(s) = 1 | − 1.51·3-s + 5·5-s − 5.13·7-s − 24.6·9-s − 31.3·11-s − 4.75·13-s − 7.58·15-s + 108.·17-s + 89.8·19-s + 7.79·21-s + 68.5·23-s + 25·25-s + 78.4·27-s − 16.5·29-s − 300.·31-s + 47.5·33-s − 25.6·35-s + 327.·37-s + 7.21·39-s + 73.4·41-s − 0.836·43-s − 123.·45-s + 228.·47-s − 316.·49-s − 164.·51-s − 647.·53-s − 156.·55-s + ⋯ |
| L(s) = 1 | − 0.292·3-s + 0.447·5-s − 0.277·7-s − 0.914·9-s − 0.859·11-s − 0.101·13-s − 0.130·15-s + 1.54·17-s + 1.08·19-s + 0.0810·21-s + 0.621·23-s + 0.200·25-s + 0.559·27-s − 0.106·29-s − 1.74·31-s + 0.250·33-s − 0.124·35-s + 1.45·37-s + 0.0296·39-s + 0.279·41-s − 0.00296·43-s − 0.409·45-s + 0.708·47-s − 0.923·49-s − 0.450·51-s − 1.67·53-s − 0.384·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 5T \) |
| good | 3 | \( 1 + 1.51T + 27T^{2} \) |
| 7 | \( 1 + 5.13T + 343T^{2} \) |
| 11 | \( 1 + 31.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 4.75T + 2.19e3T^{2} \) |
| 17 | \( 1 - 108.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 89.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 68.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 16.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 300.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 327.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 73.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 0.836T + 7.95e4T^{2} \) |
| 47 | \( 1 - 228.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 647.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 753.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 290.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 801.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 767.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 48.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 451.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 976.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.20e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 559.T + 9.12e5T^{2} \) |
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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
−9.091475954691869933509822118223, −7.900508037218158813858610402740, −7.42471178703239262660187275921, −6.13762586273907892239235371984, −5.58202375489250979067800824268, −4.89655772886487115461446884299, −3.36515532581719695454863659808, −2.73728706340027659556905928228, −1.28452531515343224835072419271, 0,
1.28452531515343224835072419271, 2.73728706340027659556905928228, 3.36515532581719695454863659808, 4.89655772886487115461446884299, 5.58202375489250979067800824268, 6.13762586273907892239235371984, 7.42471178703239262660187275921, 7.900508037218158813858610402740, 9.091475954691869933509822118223
