| L(s) = 1 | + 8·3-s + 5·5-s − 7·7-s + 37·9-s − 68·11-s − 34·13-s + 40·15-s + 74·17-s + 128·19-s − 56·21-s − 80·23-s + 25·25-s + 80·27-s − 286·29-s − 24·31-s − 544·33-s − 35·35-s − 294·37-s − 272·39-s + 66·41-s + 124·43-s + 185·45-s + 312·47-s + 49·49-s + 592·51-s + 34·53-s − 340·55-s + ⋯ |
| L(s) = 1 | + 1.53·3-s + 0.447·5-s − 0.377·7-s + 1.37·9-s − 1.86·11-s − 0.725·13-s + 0.688·15-s + 1.05·17-s + 1.54·19-s − 0.581·21-s − 0.725·23-s + 1/5·25-s + 0.570·27-s − 1.83·29-s − 0.139·31-s − 2.86·33-s − 0.169·35-s − 1.30·37-s − 1.11·39-s + 0.251·41-s + 0.439·43-s + 0.612·45-s + 0.968·47-s + 1/7·49-s + 1.62·51-s + 0.0881·53-s − 0.833·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 7 | \( 1 + p T \) |
| good | 3 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 + 68 T + p^{3} T^{2} \) |
| 13 | \( 1 + 34 T + p^{3} T^{2} \) |
| 17 | \( 1 - 74 T + p^{3} T^{2} \) |
| 19 | \( 1 - 128 T + p^{3} T^{2} \) |
| 23 | \( 1 + 80 T + p^{3} T^{2} \) |
| 29 | \( 1 + 286 T + p^{3} T^{2} \) |
| 31 | \( 1 + 24 T + p^{3} T^{2} \) |
| 37 | \( 1 + 294 T + p^{3} T^{2} \) |
| 41 | \( 1 - 66 T + p^{3} T^{2} \) |
| 43 | \( 1 - 124 T + p^{3} T^{2} \) |
| 47 | \( 1 - 312 T + p^{3} T^{2} \) |
| 53 | \( 1 - 34 T + p^{3} T^{2} \) |
| 59 | \( 1 + 168 T + p^{3} T^{2} \) |
| 61 | \( 1 + 170 T + p^{3} T^{2} \) |
| 67 | \( 1 + 564 T + p^{3} T^{2} \) |
| 71 | \( 1 - 616 T + p^{3} T^{2} \) |
| 73 | \( 1 - 250 T + p^{3} T^{2} \) |
| 79 | \( 1 + 944 T + p^{3} T^{2} \) |
| 83 | \( 1 + 672 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1430 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1270 T + p^{3} T^{2} \) |
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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
−8.218957402943854918271208482970, −7.51792246028095551918767143057, −7.29187922072324019566840825504, −5.64552504915732593979471074277, −5.30122625500647601396660387654, −3.94210211809976414676366487110, −3.05263468667901667803369116938, −2.57735105209395812517161924291, −1.59563713662409781461215843494, 0,
1.59563713662409781461215843494, 2.57735105209395812517161924291, 3.05263468667901667803369116938, 3.94210211809976414676366487110, 5.30122625500647601396660387654, 5.64552504915732593979471074277, 7.29187922072324019566840825504, 7.51792246028095551918767143057, 8.218957402943854918271208482970
