| L(s) = 1 | + 3-s − 5-s + 2.13·7-s + 9-s − 3.64·11-s + 4.85·13-s − 15-s + 1.80·17-s + 4.72·19-s + 2.13·21-s + 6.38·23-s + 25-s + 27-s − 4.65·29-s + 31-s − 3.64·33-s − 2.13·35-s − 0.588·37-s + 4.85·39-s − 8.90·41-s − 6.18·43-s − 45-s − 2.73·47-s − 2.45·49-s + 1.80·51-s + 10.9·53-s + 3.64·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.805·7-s + 0.333·9-s − 1.09·11-s + 1.34·13-s − 0.258·15-s + 0.438·17-s + 1.08·19-s + 0.465·21-s + 1.33·23-s + 0.200·25-s + 0.192·27-s − 0.865·29-s + 0.179·31-s − 0.634·33-s − 0.360·35-s − 0.0967·37-s + 0.777·39-s − 1.39·41-s − 0.943·43-s − 0.149·45-s − 0.398·47-s − 0.350·49-s + 0.252·51-s + 1.50·53-s + 0.491·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.556406934\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.556406934\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 - 2.13T + 7T^{2} \) |
| 11 | \( 1 + 3.64T + 11T^{2} \) |
| 13 | \( 1 - 4.85T + 13T^{2} \) |
| 17 | \( 1 - 1.80T + 17T^{2} \) |
| 19 | \( 1 - 4.72T + 19T^{2} \) |
| 23 | \( 1 - 6.38T + 23T^{2} \) |
| 29 | \( 1 + 4.65T + 29T^{2} \) |
| 37 | \( 1 + 0.588T + 37T^{2} \) |
| 41 | \( 1 + 8.90T + 41T^{2} \) |
| 43 | \( 1 + 6.18T + 43T^{2} \) |
| 47 | \( 1 + 2.73T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 + 5.58T + 59T^{2} \) |
| 61 | \( 1 - 4.92T + 61T^{2} \) |
| 67 | \( 1 - 13.9T + 67T^{2} \) |
| 71 | \( 1 + 14.5T + 71T^{2} \) |
| 73 | \( 1 + 2.67T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 - 0.747T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
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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
−8.536920038059165595837756183997, −7.76149441529909470699388961136, −7.37194813715153272129181357161, −6.35387503379196649612714015492, −5.25674036723347612152373670844, −4.88807300046132645981146767825, −3.61659356244709297057449253473, −3.20046448552534301009969322038, −1.97948661865930659740241934556, −0.954605222188967690795453244787,
0.954605222188967690795453244787, 1.97948661865930659740241934556, 3.20046448552534301009969322038, 3.61659356244709297057449253473, 4.88807300046132645981146767825, 5.25674036723347612152373670844, 6.35387503379196649612714015492, 7.37194813715153272129181357161, 7.76149441529909470699388961136, 8.536920038059165595837756183997
