| L(s) = 1 | + 2.47·3-s + 2.85·5-s + 3.80·7-s + 3.12·9-s + 2.73·11-s − 2.57·13-s + 7.06·15-s − 17-s − 2.99·19-s + 9.42·21-s + 6.20·23-s + 3.14·25-s + 0.316·27-s + 2.72·29-s − 0.170·31-s + 6.78·33-s + 10.8·35-s − 2.15·37-s − 6.36·39-s + 5.07·41-s − 1.66·43-s + 8.92·45-s − 2.30·47-s + 7.50·49-s − 2.47·51-s + 2.59·53-s + 7.82·55-s + ⋯ |
| L(s) = 1 | + 1.42·3-s + 1.27·5-s + 1.43·7-s + 1.04·9-s + 0.826·11-s − 0.713·13-s + 1.82·15-s − 0.242·17-s − 0.686·19-s + 2.05·21-s + 1.29·23-s + 0.629·25-s + 0.0608·27-s + 0.505·29-s − 0.0307·31-s + 1.18·33-s + 1.83·35-s − 0.354·37-s − 1.01·39-s + 0.793·41-s − 0.253·43-s + 1.33·45-s − 0.335·47-s + 1.07·49-s − 0.346·51-s + 0.356·53-s + 1.05·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.806755249\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.806755249\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
| good | 3 | \( 1 - 2.47T + 3T^{2} \) |
| 5 | \( 1 - 2.85T + 5T^{2} \) |
| 7 | \( 1 - 3.80T + 7T^{2} \) |
| 11 | \( 1 - 2.73T + 11T^{2} \) |
| 13 | \( 1 + 2.57T + 13T^{2} \) |
| 19 | \( 1 + 2.99T + 19T^{2} \) |
| 23 | \( 1 - 6.20T + 23T^{2} \) |
| 29 | \( 1 - 2.72T + 29T^{2} \) |
| 31 | \( 1 + 0.170T + 31T^{2} \) |
| 37 | \( 1 + 2.15T + 37T^{2} \) |
| 41 | \( 1 - 5.07T + 41T^{2} \) |
| 43 | \( 1 + 1.66T + 43T^{2} \) |
| 47 | \( 1 + 2.30T + 47T^{2} \) |
| 53 | \( 1 - 2.59T + 53T^{2} \) |
| 61 | \( 1 + 8.25T + 61T^{2} \) |
| 67 | \( 1 + 9.82T + 67T^{2} \) |
| 71 | \( 1 + 8.10T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 + 15.6T + 79T^{2} \) |
| 83 | \( 1 - 15.0T + 83T^{2} \) |
| 89 | \( 1 - 7.37T + 89T^{2} \) |
| 97 | \( 1 - 5.45T + 97T^{2} \) |
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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
−7.897177041632183946865877937768, −7.31756712312508593062387912352, −6.54807177422989280993005557678, −5.73020816284559481821582531553, −4.85215410527369303634474792320, −4.38659532057996774889157965368, −3.32137779658194155514231775998, −2.47048711479234632092162099955, −1.93442164497898310190217254588, −1.25942483947873861766743531871,
1.25942483947873861766743531871, 1.93442164497898310190217254588, 2.47048711479234632092162099955, 3.32137779658194155514231775998, 4.38659532057996774889157965368, 4.85215410527369303634474792320, 5.73020816284559481821582531553, 6.54807177422989280993005557678, 7.31756712312508593062387912352, 7.897177041632183946865877937768
