| L(s) = 1 | + 0.160·3-s − 5·5-s − 2.26·7-s − 26.9·9-s − 8.80·11-s + 37.4·13-s − 0.801·15-s + 71.0·17-s + 106.·19-s − 0.362·21-s − 23·23-s + 25·25-s − 8.65·27-s − 10.8·29-s + 42.2·31-s − 1.41·33-s + 11.3·35-s − 140.·37-s + 6.00·39-s + 1.84·41-s − 503.·43-s + 134.·45-s − 219.·47-s − 337.·49-s + 11.3·51-s − 59.4·53-s + 44.0·55-s + ⋯ |
| L(s) = 1 | + 0.0308·3-s − 0.447·5-s − 0.122·7-s − 0.999·9-s − 0.241·11-s + 0.798·13-s − 0.0137·15-s + 1.01·17-s + 1.28·19-s − 0.00376·21-s − 0.208·23-s + 0.200·25-s − 0.0616·27-s − 0.0695·29-s + 0.244·31-s − 0.00744·33-s + 0.0546·35-s − 0.625·37-s + 0.0246·39-s + 0.00703·41-s − 1.78·43-s + 0.446·45-s − 0.680·47-s − 0.985·49-s + 0.0312·51-s − 0.154·53-s + 0.107·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 0.160T + 27T^{2} \) |
| 7 | \( 1 + 2.26T + 343T^{2} \) |
| 11 | \( 1 + 8.80T + 1.33e3T^{2} \) |
| 13 | \( 1 - 37.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 71.0T + 4.91e3T^{2} \) |
| 19 | \( 1 - 106.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 10.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 42.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 140.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 1.84T + 6.89e4T^{2} \) |
| 43 | \( 1 + 503.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 219.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 59.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + 91.0T + 2.05e5T^{2} \) |
| 61 | \( 1 + 93.1T + 2.26e5T^{2} \) |
| 67 | \( 1 + 940.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 178.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 47.2T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.08e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 484.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 392.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.65e3T + 9.12e5T^{2} \) |
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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
−9.253939854279693732115502055502, −8.304174480293233280090974729571, −7.79591186889932104941632434728, −6.68370327282760759655966522843, −5.73175702462688106949581519798, −4.97465378152495280524072713397, −3.57032158720735069435598753201, −2.99590893938077346615988920897, −1.37153286218675604085788943349, 0,
1.37153286218675604085788943349, 2.99590893938077346615988920897, 3.57032158720735069435598753201, 4.97465378152495280524072713397, 5.73175702462688106949581519798, 6.68370327282760759655966522843, 7.79591186889932104941632434728, 8.304174480293233280090974729571, 9.253939854279693732115502055502
