| L(s) = 1 | + 2.82·3-s − 5-s − 7-s + 5.00·9-s + 11-s − 0.828·13-s − 2.82·15-s + 4.82·17-s − 2.82·21-s + 4·23-s + 25-s + 5.65·27-s − 2·29-s − 1.17·31-s + 2.82·33-s + 35-s + 0.343·37-s − 2.34·39-s + 3.17·41-s + 9.65·43-s − 5.00·45-s − 2.82·47-s + 49-s + 13.6·51-s − 7.65·53-s − 55-s + 1.17·59-s + ⋯ |
| L(s) = 1 | + 1.63·3-s − 0.447·5-s − 0.377·7-s + 1.66·9-s + 0.301·11-s − 0.229·13-s − 0.730·15-s + 1.17·17-s − 0.617·21-s + 0.834·23-s + 0.200·25-s + 1.08·27-s − 0.371·29-s − 0.210·31-s + 0.492·33-s + 0.169·35-s + 0.0564·37-s − 0.375·39-s + 0.495·41-s + 1.47·43-s − 0.745·45-s − 0.412·47-s + 0.142·49-s + 1.91·51-s − 1.05·53-s − 0.134·55-s + 0.152·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.577172574\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.577172574\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 2.82T + 3T^{2} \) |
| 13 | \( 1 + 0.828T + 13T^{2} \) |
| 17 | \( 1 - 4.82T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 37 | \( 1 - 0.343T + 37T^{2} \) |
| 41 | \( 1 - 3.17T + 41T^{2} \) |
| 43 | \( 1 - 9.65T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + 7.65T + 53T^{2} \) |
| 59 | \( 1 - 1.17T + 59T^{2} \) |
| 61 | \( 1 - 8.82T + 61T^{2} \) |
| 67 | \( 1 - 5.65T + 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 + 8.82T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 5.65T + 83T^{2} \) |
| 89 | \( 1 - 5.31T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−8.022813675981981209254701483117, −7.52910573478590516383473400161, −6.97192237442331320853536796159, −6.00709493975579705325297693642, −5.03669621774929930212642547482, −4.11643560545996901050032310893, −3.48187291089578236478131044922, −2.93069389027776955844902958256, −2.06020887469434310903254983011, −0.939476954827322869412651734936,
0.939476954827322869412651734936, 2.06020887469434310903254983011, 2.93069389027776955844902958256, 3.48187291089578236478131044922, 4.11643560545996901050032310893, 5.03669621774929930212642547482, 6.00709493975579705325297693642, 6.97192237442331320853536796159, 7.52910573478590516383473400161, 8.022813675981981209254701483117
