| L(s) = 1 | + 3.34·3-s − 5-s + 2.59·7-s + 8.19·9-s − 4.74·11-s + 6.69·13-s − 3.34·15-s − 0.748·17-s + 3.34·19-s + 8.69·21-s − 1.49·23-s + 25-s + 17.3·27-s + 3.94·29-s − 7.79·31-s − 15.8·33-s − 2.59·35-s − 37-s + 22.3·39-s − 6.44·41-s − 1.94·43-s − 8.19·45-s − 1.84·47-s − 0.251·49-s − 2.50·51-s + 10.4·53-s + 4.74·55-s + ⋯ |
| L(s) = 1 | + 1.93·3-s − 0.447·5-s + 0.981·7-s + 2.73·9-s − 1.43·11-s + 1.85·13-s − 0.863·15-s − 0.181·17-s + 0.767·19-s + 1.89·21-s − 0.312·23-s + 0.200·25-s + 3.34·27-s + 0.732·29-s − 1.39·31-s − 2.76·33-s − 0.439·35-s − 0.164·37-s + 3.58·39-s − 1.00·41-s − 0.296·43-s − 1.22·45-s − 0.269·47-s − 0.0359·49-s − 0.350·51-s + 1.43·53-s + 0.640·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.149743345\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.149743345\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 - 3.34T + 3T^{2} \) |
| 7 | \( 1 - 2.59T + 7T^{2} \) |
| 11 | \( 1 + 4.74T + 11T^{2} \) |
| 13 | \( 1 - 6.69T + 13T^{2} \) |
| 17 | \( 1 + 0.748T + 17T^{2} \) |
| 19 | \( 1 - 3.34T + 19T^{2} \) |
| 23 | \( 1 + 1.49T + 23T^{2} \) |
| 29 | \( 1 - 3.94T + 29T^{2} \) |
| 31 | \( 1 + 7.79T + 31T^{2} \) |
| 41 | \( 1 + 6.44T + 41T^{2} \) |
| 43 | \( 1 + 1.94T + 43T^{2} \) |
| 47 | \( 1 + 1.84T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 5.84T + 59T^{2} \) |
| 61 | \( 1 - 7.94T + 61T^{2} \) |
| 67 | \( 1 + 1.84T + 67T^{2} \) |
| 71 | \( 1 - 3.88T + 71T^{2} \) |
| 73 | \( 1 + 7.49T + 73T^{2} \) |
| 79 | \( 1 - 16.5T + 79T^{2} \) |
| 83 | \( 1 + 15.2T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−8.394110649933195712082926148281, −8.278690919413952983328578254942, −7.57268605922585521371378856757, −6.86071968817547910821879254241, −5.51724865811946861676360263970, −4.62596051002762559569143569225, −3.74315767723857054453348570553, −3.17344639664564008140864476561, −2.17254553428172673548696839710, −1.30296297123430177828568643884,
1.30296297123430177828568643884, 2.17254553428172673548696839710, 3.17344639664564008140864476561, 3.74315767723857054453348570553, 4.62596051002762559569143569225, 5.51724865811946861676360263970, 6.86071968817547910821879254241, 7.57268605922585521371378856757, 8.278690919413952983328578254942, 8.394110649933195712082926148281
