| L(s) = 1 | − i·3-s − i·5-s + 3.79·7-s − 9-s + 3.69·11-s + 2.60·13-s − 15-s + 1.60i·17-s − 8.20·19-s − 3.79i·21-s + (1.19 − 4.64i)23-s − 25-s + i·27-s + 0.706·29-s − 6.46i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.447i·5-s + 1.43·7-s − 0.333·9-s + 1.11·11-s + 0.723·13-s − 0.258·15-s + 0.388i·17-s − 1.88·19-s − 0.829i·21-s + (0.249 − 0.968i)23-s − 0.200·25-s + 0.192i·27-s + 0.131·29-s − 1.16i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.249 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.269044273\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.269044273\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + iT \) |
| 23 | \( 1 + (-1.19 + 4.64i)T \) |
| good | 7 | \( 1 - 3.79T + 7T^{2} \) |
| 11 | \( 1 - 3.69T + 11T^{2} \) |
| 13 | \( 1 - 2.60T + 13T^{2} \) |
| 17 | \( 1 - 1.60iT - 17T^{2} \) |
| 19 | \( 1 + 8.20T + 19T^{2} \) |
| 29 | \( 1 - 0.706T + 29T^{2} \) |
| 31 | \( 1 + 6.46iT - 31T^{2} \) |
| 37 | \( 1 + 5.01iT - 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 + 5.94T + 43T^{2} \) |
| 47 | \( 1 + 4.46iT - 47T^{2} \) |
| 53 | \( 1 + 7.25iT - 53T^{2} \) |
| 59 | \( 1 - 6.04iT - 59T^{2} \) |
| 61 | \( 1 - 3.36iT - 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 + 8.69iT - 71T^{2} \) |
| 73 | \( 1 + 0.516T + 73T^{2} \) |
| 79 | \( 1 + 3.23T + 79T^{2} \) |
| 83 | \( 1 - 4.10T + 83T^{2} \) |
| 89 | \( 1 + 5.11iT - 89T^{2} \) |
| 97 | \( 1 + 19.4iT - 97T^{2} \) |
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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
−8.230659658754483007395072597142, −7.22226426243162156822825372949, −6.48029290055013055269231015898, −5.94900942146135867276146290559, −4.94390783066946251634756833216, −4.32560097046219386458048776347, −3.61443990703397142327648121470, −2.08496570613157804422790367258, −1.72527649368548237948332824704, −0.60107779288901273858120843300,
1.28918832308562768114402928183, 2.03247788432680003179658505122, 3.27051648769534573512868575613, 3.94901865403231194859343604294, 4.71886127425772665194004753835, 5.28557729996486979060617039547, 6.37423974790004290578871918206, 6.75170676670026185200703895593, 7.80030872204992059029832479276, 8.509959955316904778784292046486
