| L(s) = 1 | + 1.39·3-s + 1.13·5-s + 0.122·7-s − 1.05·9-s + 1.04·11-s + 1.87·13-s + 1.58·15-s − 4.65·17-s − 4.43·19-s + 0.170·21-s − 5.95·23-s − 3.71·25-s − 5.65·27-s − 4.89·29-s − 2.03·31-s + 1.45·33-s + 0.138·35-s − 5.04·37-s + 2.60·39-s + 10.8·41-s − 1.42·43-s − 1.20·45-s − 8.58·47-s − 6.98·49-s − 6.48·51-s − 1.94·53-s + 1.18·55-s + ⋯ |
| L(s) = 1 | + 0.804·3-s + 0.507·5-s + 0.0462·7-s − 0.352·9-s + 0.314·11-s + 0.518·13-s + 0.408·15-s − 1.12·17-s − 1.01·19-s + 0.0371·21-s − 1.24·23-s − 0.742·25-s − 1.08·27-s − 0.909·29-s − 0.365·31-s + 0.252·33-s + 0.0234·35-s − 0.829·37-s + 0.417·39-s + 1.69·41-s − 0.217·43-s − 0.179·45-s − 1.25·47-s − 0.997·49-s − 0.907·51-s − 0.267·53-s + 0.159·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 269 | \( 1 + T \) |
| good | 3 | \( 1 - 1.39T + 3T^{2} \) |
| 5 | \( 1 - 1.13T + 5T^{2} \) |
| 7 | \( 1 - 0.122T + 7T^{2} \) |
| 11 | \( 1 - 1.04T + 11T^{2} \) |
| 13 | \( 1 - 1.87T + 13T^{2} \) |
| 17 | \( 1 + 4.65T + 17T^{2} \) |
| 19 | \( 1 + 4.43T + 19T^{2} \) |
| 23 | \( 1 + 5.95T + 23T^{2} \) |
| 29 | \( 1 + 4.89T + 29T^{2} \) |
| 31 | \( 1 + 2.03T + 31T^{2} \) |
| 37 | \( 1 + 5.04T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 1.42T + 43T^{2} \) |
| 47 | \( 1 + 8.58T + 47T^{2} \) |
| 53 | \( 1 + 1.94T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 - 5.47T + 61T^{2} \) |
| 67 | \( 1 + 7.37T + 67T^{2} \) |
| 71 | \( 1 - 5.54T + 71T^{2} \) |
| 73 | \( 1 - 9.82T + 73T^{2} \) |
| 79 | \( 1 - 6.01T + 79T^{2} \) |
| 83 | \( 1 - 8.42T + 83T^{2} \) |
| 89 | \( 1 - 1.48T + 89T^{2} \) |
| 97 | \( 1 - 1.99T + 97T^{2} \) |
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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
−8.119605213142832267197526484703, −7.48079628722363559102276454466, −6.24917647625324350732725807115, −6.15413749882031133590867248805, −4.98714384577896301644551623807, −4.03092270233291571272096262871, −3.42960782874919352302498067565, −2.23467820946110499880276457755, −1.84206838083076597171661082562, 0,
1.84206838083076597171661082562, 2.23467820946110499880276457755, 3.42960782874919352302498067565, 4.03092270233291571272096262871, 4.98714384577896301644551623807, 6.15413749882031133590867248805, 6.24917647625324350732725807115, 7.48079628722363559102276454466, 8.119605213142832267197526484703
