| L(s) = 1 | + 0.414i·2-s + 0.414i·3-s + 1.82·4-s − 0.171·6-s + 2.82i·7-s + 1.58i·8-s + 2.82·9-s + 2.41·11-s + 0.757i·12-s − 1.82i·13-s − 1.17·14-s + 3·16-s − 4.82i·17-s + 1.17i·18-s − 6·19-s + ⋯ |
| L(s) = 1 | + 0.292i·2-s + 0.239i·3-s + 0.914·4-s − 0.0700·6-s + 1.06i·7-s + 0.560i·8-s + 0.942·9-s + 0.727·11-s + 0.218i·12-s − 0.507i·13-s − 0.313·14-s + 0.750·16-s − 1.17i·17-s + 0.276i·18-s − 1.37·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.74268 + 1.07703i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.74268 + 1.07703i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 0.414iT - 2T^{2} \) |
| 3 | \( 1 - 0.414iT - 3T^{2} \) |
| 7 | \( 1 - 2.82iT - 7T^{2} \) |
| 11 | \( 1 - 2.41T + 11T^{2} \) |
| 13 | \( 1 + 1.82iT - 13T^{2} \) |
| 17 | \( 1 + 4.82iT - 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 7.65iT - 23T^{2} \) |
| 31 | \( 1 + 4.07T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 - 12.4T + 41T^{2} \) |
| 43 | \( 1 + 6.41iT - 43T^{2} \) |
| 47 | \( 1 - 5.24iT - 47T^{2} \) |
| 53 | \( 1 - 7.48iT - 53T^{2} \) |
| 59 | \( 1 + 7.65T + 59T^{2} \) |
| 61 | \( 1 - 0.828T + 61T^{2} \) |
| 67 | \( 1 + 5.65iT - 67T^{2} \) |
| 71 | \( 1 + 3.17T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 0.414T + 79T^{2} \) |
| 83 | \( 1 - 3.65iT - 83T^{2} \) |
| 89 | \( 1 + 4.48T + 89T^{2} \) |
| 97 | \( 1 + 12.4iT - 97T^{2} \) |
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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
−10.69730415406128022271304206491, −9.528321935720974650260012285855, −8.996168147526799490040821308719, −7.73591246461254823243557868417, −7.11944565241574431154064201003, −6.08437743879278583693283662347, −5.39009931274938719295779736031, −4.10145923550051239354910802045, −2.81766007168733967353649384265, −1.70564469692638566916596330688,
1.20956556050702346931000931835, 2.22774949831765736529290230890, 3.85610452203517744381777545836, 4.35492792402716654153039549420, 6.25678719609871240587496123779, 6.66324152571396465483351384904, 7.47657781401479545718473787648, 8.427815578746243609355335842926, 9.628007061351221685640068391345, 10.56918776066708502731329690837
