| L(s) = 1 | − 1.60i·5-s + (2.41 + 1.08i)7-s − 4.20·11-s − 3.26·13-s − 0.666i·17-s + (2.31 + 3.69i)19-s − 1.74·23-s + 2.41·25-s − 8.97i·29-s − 7.89·31-s + (1.74 − 3.88i)35-s + 8.54i·37-s + 9.71·41-s − 0.242·43-s + 11.6i·47-s + ⋯ |
| L(s) = 1 | − 0.719i·5-s + (0.912 + 0.409i)7-s − 1.26·11-s − 0.906·13-s − 0.161i·17-s + (0.530 + 0.847i)19-s − 0.362·23-s + 0.482·25-s − 1.66i·29-s − 1.41·31-s + (0.294 − 0.656i)35-s + 1.40i·37-s + 1.51·41-s − 0.0370·43-s + 1.69i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.137 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.137 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.008750907\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.008750907\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.41 - 1.08i)T \) |
| 19 | \( 1 + (-2.31 - 3.69i)T \) |
| good | 5 | \( 1 + 1.60iT - 5T^{2} \) |
| 11 | \( 1 + 4.20T + 11T^{2} \) |
| 13 | \( 1 + 3.26T + 13T^{2} \) |
| 17 | \( 1 + 0.666iT - 17T^{2} \) |
| 23 | \( 1 + 1.74T + 23T^{2} \) |
| 29 | \( 1 + 8.97iT - 29T^{2} \) |
| 31 | \( 1 + 7.89T + 31T^{2} \) |
| 37 | \( 1 - 8.54iT - 37T^{2} \) |
| 41 | \( 1 - 9.71T + 41T^{2} \) |
| 43 | \( 1 + 0.242T + 43T^{2} \) |
| 47 | \( 1 - 11.6iT - 47T^{2} \) |
| 53 | \( 1 - 3.71iT - 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 - 1.53iT - 61T^{2} \) |
| 67 | \( 1 + 12.0iT - 67T^{2} \) |
| 71 | \( 1 - 3.71iT - 71T^{2} \) |
| 73 | \( 1 - 9.81iT - 73T^{2} \) |
| 79 | \( 1 - 8.54iT - 79T^{2} \) |
| 83 | \( 1 - 4.82iT - 83T^{2} \) |
| 89 | \( 1 + 9.71T + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−8.213698471211672280671057461182, −7.902048941185491608986934770139, −7.34376189604844159437389968799, −6.05400134549458281036573173607, −5.48086899077289184444553471818, −4.82030125438107941548627364347, −4.26076626420212026238828710406, −2.92928093436004919070264704788, −2.20643662098708113930348153061, −1.14924543163332610712497995635,
0.27735285221846201686415877450, 1.78453907022497985214775260015, 2.61696501140074960202277761569, 3.40348961043425144356921876089, 4.47950088170877700300571902586, 5.17265609194887244328247688206, 5.70522284928985427943454732328, 7.05296785294663006649026562116, 7.24774336492750096356523543187, 7.891195071209426133466595044925
