| L(s) = 1 | + 2.41i·2-s − i·3-s − 3.82·4-s + 2.41·6-s − 1.41i·7-s − 4.41i·8-s − 9-s − 2.24·11-s + 3.82i·12-s + 3.41i·13-s + 3.41·14-s + 2.99·16-s + 1.17i·17-s − 2.41i·18-s + 19-s + ⋯ |
| L(s) = 1 | + 1.70i·2-s − 0.577i·3-s − 1.91·4-s + 0.985·6-s − 0.534i·7-s − 1.56i·8-s − 0.333·9-s − 0.676·11-s + 1.10i·12-s + 0.946i·13-s + 0.912·14-s + 0.749·16-s + 0.284i·17-s − 0.569i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6875691443\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6875691443\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 2.41iT - 2T^{2} \) |
| 7 | \( 1 + 1.41iT - 7T^{2} \) |
| 11 | \( 1 + 2.24T + 11T^{2} \) |
| 13 | \( 1 - 3.41iT - 13T^{2} \) |
| 17 | \( 1 - 1.17iT - 17T^{2} \) |
| 23 | \( 1 + 7.65iT - 23T^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 + 3.17T + 31T^{2} \) |
| 37 | \( 1 + 3.41iT - 37T^{2} \) |
| 41 | \( 1 + 0.242T + 41T^{2} \) |
| 43 | \( 1 + 12.2iT - 43T^{2} \) |
| 47 | \( 1 + 7.65iT - 47T^{2} \) |
| 53 | \( 1 + 8iT - 53T^{2} \) |
| 59 | \( 1 + 12.4T + 59T^{2} \) |
| 61 | \( 1 - 7.31T + 61T^{2} \) |
| 67 | \( 1 + 9.65iT - 67T^{2} \) |
| 71 | \( 1 + 10.8T + 71T^{2} \) |
| 73 | \( 1 - 7.65iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 12.8iT - 83T^{2} \) |
| 89 | \( 1 + 5.89T + 89T^{2} \) |
| 97 | \( 1 + 9.75iT - 97T^{2} \) |
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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
−8.961265015224522891456418352004, −8.521873842662954651546654429488, −7.58305901311162843864826296523, −7.08614972232127333827418564637, −6.41811855240031157749682284004, −5.57555938589957982824849317783, −4.73190062008191779565197100823, −3.79751027969376831171306675016, −2.14762180092936500260585940655, −0.28717566319066689024259488157,
1.35632357464632128055293118146, 2.71738885677831532616718936191, 3.18749556939005336142515482713, 4.26822578026919467300975867846, 5.17832905829897404385592135440, 5.85064324714691096137798970854, 7.49155896525733993035316402834, 8.315745563499942533534695799351, 9.322863779161367176463310088077, 9.648247029701314630311846036928
