| L(s) = 1 | − 3·3-s + 5·5-s − 3.02·7-s + 9·9-s − 28.8·11-s − 73.6·13-s − 15·15-s + 2.17·17-s − 32.8·19-s + 9.07·21-s − 23·23-s + 25·25-s − 27·27-s − 64.0·29-s + 203.·31-s + 86.4·33-s − 15.1·35-s + 37.1·37-s + 221.·39-s − 135.·41-s + 122.·43-s + 45·45-s + 206.·47-s − 333.·49-s − 6.52·51-s + 589.·53-s − 144.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 0.163·7-s + 0.333·9-s − 0.789·11-s − 1.57·13-s − 0.258·15-s + 0.0310·17-s − 0.396·19-s + 0.0942·21-s − 0.208·23-s + 0.200·25-s − 0.192·27-s − 0.410·29-s + 1.17·31-s + 0.455·33-s − 0.0730·35-s + 0.165·37-s + 0.907·39-s − 0.514·41-s + 0.434·43-s + 0.149·45-s + 0.640·47-s − 0.973·49-s − 0.0179·51-s + 1.52·53-s − 0.353·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.153059471\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.153059471\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 + 23T \) |
| good | 7 | \( 1 + 3.02T + 343T^{2} \) |
| 11 | \( 1 + 28.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 73.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 2.17T + 4.91e3T^{2} \) |
| 19 | \( 1 + 32.8T + 6.85e3T^{2} \) |
| 29 | \( 1 + 64.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 203.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 37.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 135.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 122.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 206.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 589.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 455.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 670.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 126.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 825.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 944.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 198.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 241.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 144.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.79e3T + 9.12e5T^{2} \) |
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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
−9.435944103912039654565934418734, −8.351867692686598636402126868049, −7.48025636534117671802703051980, −6.75680743170725071044078060652, −5.81159974529531376382337385905, −5.09849831316475404103109521637, −4.31146744191869635998802930469, −2.88635990404755970234019314661, −2.02991732189212107868842306619, −0.52791961495958279816005994996,
0.52791961495958279816005994996, 2.02991732189212107868842306619, 2.88635990404755970234019314661, 4.31146744191869635998802930469, 5.09849831316475404103109521637, 5.81159974529531376382337385905, 6.75680743170725071044078060652, 7.48025636534117671802703051980, 8.351867692686598636402126868049, 9.435944103912039654565934418734
