L(s) = 1 | − i·3-s + (−0.707 − 0.707i)5-s − 9-s − 1.41·11-s − i·13-s + (−0.707 + 0.707i)15-s + 1.00i·25-s + i·27-s + 1.41i·33-s − 39-s − 1.41·41-s + (0.707 + 0.707i)45-s + 1.41i·47-s − 49-s + (1.00 + 1.00i)55-s + ⋯ |
L(s) = 1 | − i·3-s + (−0.707 − 0.707i)5-s − 9-s − 1.41·11-s − i·13-s + (−0.707 + 0.707i)15-s + 1.00i·25-s + i·27-s + 1.41i·33-s − 39-s − 1.41·41-s + (0.707 + 0.707i)45-s + 1.41i·47-s − 49-s + (1.00 + 1.00i)55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2857166347\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2857166347\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (0.707 + 0.707i)T \) |
| 13 | \( 1 + iT \) |
good | 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + 1.41T + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + 1.41T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 1.41iT - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + 1.41T + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 - 1.41T + T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + 1.41iT - T^{2} \) |
| 89 | \( 1 + 1.41T + T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−8.083953224716452184605481379128, −7.88252884315716033665719419867, −7.11836012398796632583323953165, −6.10808545099996244088742012967, −5.32847254109513475673297968515, −4.74980287587774842555575546414, −3.41149041340796205780890360133, −2.70326543779234424017935027587, −1.46392472891688415151088303840, −0.16865417843099966124929236871,
2.21343661920868976287925549200, 3.10276304988602041828966018836, 3.82395598087524067638714902922, 4.69711544863249307091986314674, 5.32936288501132113424176934000, 6.36187213700215120130370121575, 7.09959561207715684127301059316, 8.020414514911455011988180395843, 8.486751014530015794514832401036, 9.451631490944719993520534895006