| L(s) = 1 | + 1.61·2-s + 0.607·4-s − 2.02·5-s − 1.29·7-s − 2.24·8-s − 3.26·10-s − 5.65·11-s − 2.24·13-s − 2.09·14-s − 4.84·16-s − 2.06·17-s − 19-s − 1.22·20-s − 9.13·22-s + 2.15·23-s − 0.901·25-s − 3.62·26-s − 0.788·28-s + 7.29·29-s + 3.56·31-s − 3.32·32-s − 3.34·34-s + 2.62·35-s + 0.880·37-s − 1.61·38-s + 4.55·40-s − 10.1·41-s + ⋯ |
| L(s) = 1 | + 1.14·2-s + 0.303·4-s − 0.905·5-s − 0.490·7-s − 0.795·8-s − 1.03·10-s − 1.70·11-s − 0.622·13-s − 0.560·14-s − 1.21·16-s − 0.501·17-s − 0.229·19-s − 0.274·20-s − 1.94·22-s + 0.450·23-s − 0.180·25-s − 0.710·26-s − 0.148·28-s + 1.35·29-s + 0.639·31-s − 0.588·32-s − 0.572·34-s + 0.444·35-s + 0.144·37-s − 0.261·38-s + 0.719·40-s − 1.58·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8338022680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8338022680\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 5 | \( 1 + 2.02T + 5T^{2} \) |
| 7 | \( 1 + 1.29T + 7T^{2} \) |
| 11 | \( 1 + 5.65T + 11T^{2} \) |
| 13 | \( 1 + 2.24T + 13T^{2} \) |
| 17 | \( 1 + 2.06T + 17T^{2} \) |
| 23 | \( 1 - 2.15T + 23T^{2} \) |
| 29 | \( 1 - 7.29T + 29T^{2} \) |
| 31 | \( 1 - 3.56T + 31T^{2} \) |
| 37 | \( 1 - 0.880T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + 8.07T + 43T^{2} \) |
| 53 | \( 1 + 8.45T + 53T^{2} \) |
| 59 | \( 1 - 8.85T + 59T^{2} \) |
| 61 | \( 1 + 0.251T + 61T^{2} \) |
| 67 | \( 1 - 6.22T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 + 8.67T + 73T^{2} \) |
| 79 | \( 1 + 5.08T + 79T^{2} \) |
| 83 | \( 1 + 7.62T + 83T^{2} \) |
| 89 | \( 1 - 7.55T + 89T^{2} \) |
| 97 | \( 1 - 17.7T + 97T^{2} \) |
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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
−7.903865988512921284891378500607, −6.88527833056881558720018297893, −6.47434393289414681471871148177, −5.47908219971018736533652222923, −4.86631935733945389650739206962, −4.49770214584012023581001673219, −3.45298521497740940243057675389, −3.00961482955529745743064607360, −2.21696049710178325382292294222, −0.35014193444037043870470443955,
0.35014193444037043870470443955, 2.21696049710178325382292294222, 3.00961482955529745743064607360, 3.45298521497740940243057675389, 4.49770214584012023581001673219, 4.86631935733945389650739206962, 5.47908219971018736533652222923, 6.47434393289414681471871148177, 6.88527833056881558720018297893, 7.903865988512921284891378500607
