| L(s) = 1 | − 1.65·3-s − 0.641·5-s − 1.04·7-s − 0.251·9-s − 3.12·11-s − 1.00·13-s + 1.06·15-s − 4.94·17-s − 8.16·19-s + 1.73·21-s + 1.05·23-s − 4.58·25-s + 5.39·27-s − 10.2·29-s + 7.21·31-s + 5.18·33-s + 0.672·35-s − 2.17·37-s + 1.66·39-s + 9.09·41-s + 1.16·43-s + 0.161·45-s + 3.67·47-s − 5.90·49-s + 8.20·51-s − 5.62·53-s + 2.00·55-s + ⋯ |
| L(s) = 1 | − 0.957·3-s − 0.287·5-s − 0.396·7-s − 0.0837·9-s − 0.943·11-s − 0.278·13-s + 0.274·15-s − 1.20·17-s − 1.87·19-s + 0.379·21-s + 0.220·23-s − 0.917·25-s + 1.03·27-s − 1.90·29-s + 1.29·31-s + 0.903·33-s + 0.113·35-s − 0.357·37-s + 0.266·39-s + 1.42·41-s + 0.177·43-s + 0.0240·45-s + 0.536·47-s − 0.843·49-s + 1.14·51-s − 0.772·53-s + 0.270·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2521425046\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2521425046\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 + 1.65T + 3T^{2} \) |
| 5 | \( 1 + 0.641T + 5T^{2} \) |
| 7 | \( 1 + 1.04T + 7T^{2} \) |
| 11 | \( 1 + 3.12T + 11T^{2} \) |
| 13 | \( 1 + 1.00T + 13T^{2} \) |
| 17 | \( 1 + 4.94T + 17T^{2} \) |
| 19 | \( 1 + 8.16T + 19T^{2} \) |
| 23 | \( 1 - 1.05T + 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 - 7.21T + 31T^{2} \) |
| 37 | \( 1 + 2.17T + 37T^{2} \) |
| 41 | \( 1 - 9.09T + 41T^{2} \) |
| 43 | \( 1 - 1.16T + 43T^{2} \) |
| 47 | \( 1 - 3.67T + 47T^{2} \) |
| 53 | \( 1 + 5.62T + 53T^{2} \) |
| 59 | \( 1 - 9.72T + 59T^{2} \) |
| 61 | \( 1 - 0.384T + 61T^{2} \) |
| 67 | \( 1 + 6.44T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 + 9.94T + 73T^{2} \) |
| 79 | \( 1 - 9.57T + 79T^{2} \) |
| 83 | \( 1 + 5.35T + 83T^{2} \) |
| 89 | \( 1 + 15.2T + 89T^{2} \) |
| 97 | \( 1 - 4.42T + 97T^{2} \) |
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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
−8.439986608366383596482944510872, −7.65986502721025239683543444692, −6.85141438253081915971167602764, −6.13231179523664019565529147520, −5.63165551156336553575953410876, −4.65080057820530365084032164103, −4.10532118692167920956541067549, −2.84965694255173139200680499069, −2.03446575473847956769867988729, −0.28033525776881926393579112587,
0.28033525776881926393579112587, 2.03446575473847956769867988729, 2.84965694255173139200680499069, 4.10532118692167920956541067549, 4.65080057820530365084032164103, 5.63165551156336553575953410876, 6.13231179523664019565529147520, 6.85141438253081915971167602764, 7.65986502721025239683543444692, 8.439986608366383596482944510872
