| L(s) = 1 | + (−0.775 − 2.09i)5-s − 2.02i·7-s − 2.79·11-s + 4.21i·13-s − 6.37i·17-s + 19-s + 5.46i·23-s + (−3.79 + 3.25i)25-s − 8.08·29-s − 0.177·31-s + (−4.23 + 1.56i)35-s + 0.574i·37-s − 2.75·41-s − 0.919i·43-s − 10.4i·47-s + ⋯ |
| L(s) = 1 | + (−0.346 − 0.937i)5-s − 0.764i·7-s − 0.843·11-s + 1.16i·13-s − 1.54i·17-s + 0.229·19-s + 1.13i·23-s + (−0.759 + 0.650i)25-s − 1.50·29-s − 0.0318·31-s + (−0.716 + 0.265i)35-s + 0.0944i·37-s − 0.429·41-s − 0.140i·43-s − 1.51i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.346 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.346 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2667939699\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2667939699\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.775 + 2.09i)T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 2.02iT - 7T^{2} \) |
| 11 | \( 1 + 2.79T + 11T^{2} \) |
| 13 | \( 1 - 4.21iT - 13T^{2} \) |
| 17 | \( 1 + 6.37iT - 17T^{2} \) |
| 23 | \( 1 - 5.46iT - 23T^{2} \) |
| 29 | \( 1 + 8.08T + 29T^{2} \) |
| 31 | \( 1 + 0.177T + 31T^{2} \) |
| 37 | \( 1 - 0.574iT - 37T^{2} \) |
| 41 | \( 1 + 2.75T + 41T^{2} \) |
| 43 | \( 1 + 0.919iT - 43T^{2} \) |
| 47 | \( 1 + 10.4iT - 47T^{2} \) |
| 53 | \( 1 - 5.86iT - 53T^{2} \) |
| 59 | \( 1 + 5.10T + 59T^{2} \) |
| 61 | \( 1 - 0.314T + 61T^{2} \) |
| 67 | \( 1 - 5.10iT - 67T^{2} \) |
| 71 | \( 1 + 0.747T + 71T^{2} \) |
| 73 | \( 1 - 5.52iT - 73T^{2} \) |
| 79 | \( 1 - 4.70T + 79T^{2} \) |
| 83 | \( 1 - 10.9iT - 83T^{2} \) |
| 89 | \( 1 - 2.97T + 89T^{2} \) |
| 97 | \( 1 - 6.71iT - 97T^{2} \) |
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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
−8.996027025596633339805599159514, −7.937544201995876874784006085750, −7.41142832638409340968894317207, −6.84609682983469941748119746344, −5.53020762742260745819432473455, −5.08164041998511116828683397497, −4.19842798939998363923538052155, −3.51384000260677081039935277810, −2.23976538592419237229211747591, −1.12694743484226437179769062898,
0.084439666035444145038061560743, 1.91108777303249248388338149398, 2.80939387390381738788624046758, 3.46431045434601477900917835804, 4.48329934115877437076879189663, 5.59319934085940549105543007247, 5.98658749698720476923282733681, 6.88833956652069101959899626095, 7.88487228815382380224411807311, 8.087366861701538327815983959665
