| L(s) = 1 | + 3·3-s + 5·5-s − 35.1·7-s + 9·9-s + 13.7·11-s + 13·13-s + 15·15-s + 87.1·17-s + 11.6·19-s − 105.·21-s − 108.·23-s + 25·25-s + 27·27-s − 150.·29-s − 128.·31-s + 41.2·33-s − 175.·35-s − 11.7·37-s + 39·39-s + 366.·41-s + 301.·43-s + 45·45-s − 156.·47-s + 892.·49-s + 261.·51-s − 229.·53-s + 68.7·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.89·7-s + 0.333·9-s + 0.376·11-s + 0.277·13-s + 0.258·15-s + 1.24·17-s + 0.141·19-s − 1.09·21-s − 0.981·23-s + 0.200·25-s + 0.192·27-s − 0.964·29-s − 0.742·31-s + 0.217·33-s − 0.848·35-s − 0.0523·37-s + 0.160·39-s + 1.39·41-s + 1.06·43-s + 0.149·45-s − 0.486·47-s + 2.60·49-s + 0.717·51-s − 0.593·53-s + 0.168·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 - 5T \) |
| 13 | \( 1 - 13T \) |
| good | 7 | \( 1 + 35.1T + 343T^{2} \) |
| 11 | \( 1 - 13.7T + 1.33e3T^{2} \) |
| 17 | \( 1 - 87.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 11.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 108.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 150.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 128.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 11.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 366.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 301.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 156.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 229.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 377.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 406.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 352.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 842.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 50.4T + 3.89e5T^{2} \) |
| 79 | \( 1 + 695.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 675.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 943.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 73.8T + 9.12e5T^{2} \) |
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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
−8.983770738720466213523883455100, −7.77914174161419444044131370657, −7.11685355185143371520641982527, −6.08772275309199311849767752079, −5.74327854640301010346055229222, −4.15458750426137820473979509498, −3.41179852419118532223301766889, −2.68310646449846113402485127350, −1.39068828216685182347203874719, 0,
1.39068828216685182347203874719, 2.68310646449846113402485127350, 3.41179852419118532223301766889, 4.15458750426137820473979509498, 5.74327854640301010346055229222, 6.08772275309199311849767752079, 7.11685355185143371520641982527, 7.77914174161419444044131370657, 8.983770738720466213523883455100
