L(s) = 1 | + (0.707 + 0.707i)3-s + (0.707 + 0.707i)5-s + (−0.707 + 0.707i)7-s + 1.00i·9-s + 1.00i·15-s + (−1.41 − 1.41i)17-s − 1.41·19-s − 1.00·21-s + (1 + i)23-s + 1.00i·25-s + (−0.707 + 0.707i)27-s + 1.41i·31-s − 1.00·35-s + (1 + i)37-s + 1.41·41-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)3-s + (0.707 + 0.707i)5-s + (−0.707 + 0.707i)7-s + 1.00i·9-s + 1.00i·15-s + (−1.41 − 1.41i)17-s − 1.41·19-s − 1.00·21-s + (1 + i)23-s + 1.00i·25-s + (−0.707 + 0.707i)27-s + 1.41i·31-s − 1.00·35-s + (1 + i)37-s + 1.41·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.475945180\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.475945180\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 + (1.41 + 1.41i)T + iT^{2} \) |
| 19 | \( 1 + 1.41T + T^{2} \) |
| 23 | \( 1 + (-1 - i)T + iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 - 1.41iT - T^{2} \) |
| 37 | \( 1 + (-1 - i)T + iT^{2} \) |
| 41 | \( 1 - 1.41T + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 + 2iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + 1.41iT - T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.152762337271154083849193637719, −8.647188616933480927540422401622, −7.51261322385210389200016227239, −6.77285201365310256102065864059, −6.12922659319171642054425028512, −5.15444193027023143104803891148, −4.47406885138366541483869650523, −3.26446227484495128333845277901, −2.74538573981288335019967272345, −1.99740591748223141957097562317,
0.75864946296297369517950204106, 2.03515831636750882701375497583, 2.62247524080756529876846945993, 4.10377789202258324898164171755, 4.29412392941798015255737200479, 5.87555554137069728187084209125, 6.35295122920641661242240613270, 6.95052263867316841114402604751, 7.939208453326992675755090436258, 8.608964089648895358324878786181