| L(s) = 1 | − 2.56·3-s + 0.330·5-s − 0.779·7-s + 3.55·9-s − 5.15·11-s + 5.72·13-s − 0.847·15-s − 17-s − 2.03·19-s + 1.99·21-s − 2.48·23-s − 4.89·25-s − 1.43·27-s − 7.11·29-s − 7.80·31-s + 13.2·33-s − 0.257·35-s + 2.75·37-s − 14.6·39-s + 3.44·41-s + 7.75·43-s + 1.17·45-s + 6.76·47-s − 6.39·49-s + 2.56·51-s + 11.0·53-s − 1.70·55-s + ⋯ |
| L(s) = 1 | − 1.47·3-s + 0.147·5-s − 0.294·7-s + 1.18·9-s − 1.55·11-s + 1.58·13-s − 0.218·15-s − 0.242·17-s − 0.466·19-s + 0.435·21-s − 0.517·23-s − 0.978·25-s − 0.275·27-s − 1.32·29-s − 1.40·31-s + 2.29·33-s − 0.0435·35-s + 0.453·37-s − 2.34·39-s + 0.537·41-s + 1.18·43-s + 0.175·45-s + 0.986·47-s − 0.913·49-s + 0.358·51-s + 1.51·53-s − 0.229·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5288597216\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5288597216\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 + 2.56T + 3T^{2} \) |
| 5 | \( 1 - 0.330T + 5T^{2} \) |
| 7 | \( 1 + 0.779T + 7T^{2} \) |
| 11 | \( 1 + 5.15T + 11T^{2} \) |
| 13 | \( 1 - 5.72T + 13T^{2} \) |
| 19 | \( 1 + 2.03T + 19T^{2} \) |
| 23 | \( 1 + 2.48T + 23T^{2} \) |
| 29 | \( 1 + 7.11T + 29T^{2} \) |
| 31 | \( 1 + 7.80T + 31T^{2} \) |
| 37 | \( 1 - 2.75T + 37T^{2} \) |
| 41 | \( 1 - 3.44T + 41T^{2} \) |
| 43 | \( 1 - 7.75T + 43T^{2} \) |
| 47 | \( 1 - 6.76T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 61 | \( 1 + 1.33T + 61T^{2} \) |
| 67 | \( 1 + 4.09T + 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 - 7.56T + 73T^{2} \) |
| 79 | \( 1 + 5.69T + 79T^{2} \) |
| 83 | \( 1 + 6.36T + 83T^{2} \) |
| 89 | \( 1 + 1.57T + 89T^{2} \) |
| 97 | \( 1 + 1.93T + 97T^{2} \) |
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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
−7.65435257300867232356144247355, −7.09800897666786115964227236154, −6.07199163954468516215313552060, −5.82356816231892371411126127116, −5.37752647977959931388591539912, −4.32649648740805531867037865079, −3.74829193491256503807254497026, −2.57774955023234108119322518853, −1.59937290305522361360682621621, −0.38804043004400008170297658625,
0.38804043004400008170297658625, 1.59937290305522361360682621621, 2.57774955023234108119322518853, 3.74829193491256503807254497026, 4.32649648740805531867037865079, 5.37752647977959931388591539912, 5.82356816231892371411126127116, 6.07199163954468516215313552060, 7.09800897666786115964227236154, 7.65435257300867232356144247355
