| L(s) = 1 | + 1.96i·3-s + 3.88i·5-s + (−0.0756 − 2.64i)7-s − 0.873·9-s + (−3.30 + 0.323i)11-s + 13-s − 7.64·15-s + 1.57·17-s − 5.41·19-s + (5.20 − 0.148i)21-s + 0.122·23-s − 10.1·25-s + 4.18i·27-s + 6.84i·29-s − 8.22i·31-s + ⋯ |
| L(s) = 1 | + 1.13i·3-s + 1.73i·5-s + (−0.0285 − 0.999i)7-s − 0.291·9-s + (−0.995 + 0.0974i)11-s + 0.277·13-s − 1.97·15-s + 0.381·17-s − 1.24·19-s + (1.13 − 0.0324i)21-s + 0.0255·23-s − 2.02·25-s + 0.805i·27-s + 1.27i·29-s − 1.47i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.125 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.125 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06919305708\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06919305708\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.0756 + 2.64i)T \) |
| 11 | \( 1 + (3.30 - 0.323i)T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 1.96iT - 3T^{2} \) |
| 5 | \( 1 - 3.88iT - 5T^{2} \) |
| 17 | \( 1 - 1.57T + 17T^{2} \) |
| 19 | \( 1 + 5.41T + 19T^{2} \) |
| 23 | \( 1 - 0.122T + 23T^{2} \) |
| 29 | \( 1 - 6.84iT - 29T^{2} \) |
| 31 | \( 1 + 8.22iT - 31T^{2} \) |
| 37 | \( 1 + 7.64T + 37T^{2} \) |
| 41 | \( 1 - 2.15T + 41T^{2} \) |
| 43 | \( 1 + 10.9iT - 43T^{2} \) |
| 47 | \( 1 + 4.95iT - 47T^{2} \) |
| 53 | \( 1 - 5.36T + 53T^{2} \) |
| 59 | \( 1 + 0.921iT - 59T^{2} \) |
| 61 | \( 1 + 3.44T + 61T^{2} \) |
| 67 | \( 1 + 8.27T + 67T^{2} \) |
| 71 | \( 1 + 8.06T + 71T^{2} \) |
| 73 | \( 1 + 0.839T + 73T^{2} \) |
| 79 | \( 1 + 6.45iT - 79T^{2} \) |
| 83 | \( 1 + 2.23T + 83T^{2} \) |
| 89 | \( 1 - 6.64iT - 89T^{2} \) |
| 97 | \( 1 - 2.57iT - 97T^{2} \) |
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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
−9.159405303928132985521976627788, −8.236479743099473233264505104398, −7.29788133443561065211772704719, −7.03129956129699446166810286025, −6.07223862426287424578336706191, −5.23665981887864357894748835137, −4.24452023906196484833000344984, −3.67765005039313449281370883080, −3.01509014066393043616376750878, −1.98699985203390181681013732360,
0.01953273872343010182418571061, 1.22302919990374154721064457754, 1.94974030165615674405304041466, 2.87953457043817854301580789155, 4.29131422292095448681607938473, 4.95124063135169749053119045763, 5.75199820866656056355528546664, 6.23469730268167919601071277637, 7.29557737887815267794758042116, 8.164136238861306198011367398201
