| L(s) = 1 | − 0.327i·3-s − i·5-s + 2.33·7-s + 2.89·9-s + 3.77·11-s + 1.12·13-s − 0.327·15-s + 2.52i·17-s + 3.12·19-s − 0.765i·21-s + (−1.12 + 4.66i)23-s − 25-s − 1.92i·27-s − 4.41·29-s − 2.01i·31-s + ⋯ |
| L(s) = 1 | − 0.188i·3-s − 0.447i·5-s + 0.884·7-s + 0.964·9-s + 1.13·11-s + 0.312·13-s − 0.0844·15-s + 0.612i·17-s + 0.716·19-s − 0.166i·21-s + (−0.235 + 0.971i)23-s − 0.200·25-s − 0.370i·27-s − 0.820·29-s − 0.361i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.327306262\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.327306262\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (1.12 - 4.66i)T \) |
| good | 3 | \( 1 + 0.327iT - 3T^{2} \) |
| 7 | \( 1 - 2.33T + 7T^{2} \) |
| 11 | \( 1 - 3.77T + 11T^{2} \) |
| 13 | \( 1 - 1.12T + 13T^{2} \) |
| 17 | \( 1 - 2.52iT - 17T^{2} \) |
| 19 | \( 1 - 3.12T + 19T^{2} \) |
| 29 | \( 1 + 4.41T + 29T^{2} \) |
| 31 | \( 1 + 2.01iT - 31T^{2} \) |
| 37 | \( 1 - 7.54iT - 37T^{2} \) |
| 41 | \( 1 + 4.88T + 41T^{2} \) |
| 43 | \( 1 - 6.20T + 43T^{2} \) |
| 47 | \( 1 - 7.22iT - 47T^{2} \) |
| 53 | \( 1 + 4.78iT - 53T^{2} \) |
| 59 | \( 1 + 9.02iT - 59T^{2} \) |
| 61 | \( 1 - 3.02iT - 61T^{2} \) |
| 67 | \( 1 - 8.58T + 67T^{2} \) |
| 71 | \( 1 + 9.56iT - 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 - 3.12T + 79T^{2} \) |
| 83 | \( 1 - 4.80T + 83T^{2} \) |
| 89 | \( 1 - 17.8iT - 89T^{2} \) |
| 97 | \( 1 + 8.55iT - 97T^{2} \) |
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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
−9.339411540848028544844480309975, −8.310765308251130099140114675516, −7.75060793982071878721969630502, −6.89606196190133829230817942101, −6.04164390460124222571615343439, −5.08751118183545062775637050000, −4.25847802276020531215912618057, −3.51240832579943985136815715465, −1.79055291497773954120167581946, −1.23704051509131782844354866514,
1.13708736061798995791539727012, 2.18795775486908141635890826488, 3.55986055820453425261299709256, 4.26472752208974473948535434166, 5.13059627031148528373462203549, 6.13453606203313815892760513543, 7.06820398267337913480303957240, 7.51696543999673691220791733626, 8.603831972490699860706834542605, 9.265320378120619458619755806336
