| L(s) = 1 | − i·7-s − 2·11-s − 4i·13-s − 2i·17-s + 2·19-s + 4i·23-s − 2·29-s − 6·31-s − 6i·37-s − 6·41-s + 4i·43-s − 49-s + 8i·53-s − 10·61-s − 12i·67-s + ⋯ |
| L(s) = 1 | − 0.377i·7-s − 0.603·11-s − 1.10i·13-s − 0.485i·17-s + 0.458·19-s + 0.834i·23-s − 0.371·29-s − 1.07·31-s − 0.986i·37-s − 0.937·41-s + 0.609i·43-s − 0.142·49-s + 1.09i·53-s − 1.28·61-s − 1.46i·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 8iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 - 14T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 - 8iT - 97T^{2} \) |
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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
−7.71365334134835009778398683011, −7.09415911515337021282984537498, −6.15694733387371416844468269218, −5.37383764533850224516432471381, −4.96433606679176343506685122142, −3.80413663548048326674602509337, −3.22302511162094648131100349120, −2.29920768027862570972316195936, −1.15363528362704499902176387118, 0,
1.53983649923202957052397479781, 2.31064238112607934559746006700, 3.26693954765917893018972572389, 4.08915582045239615683526708959, 4.92082474570975254954751338023, 5.54875833720104528034532968149, 6.39608377613032801378874977343, 6.98510792158055500563968383464, 7.74364577936817903234453589360
