L(s) = 1 | + (−0.707 − 0.707i)3-s + 1.00i·9-s + 1.41i·11-s + 2i·17-s − 1.41·19-s + 25-s + (0.707 − 0.707i)27-s + (1.00 − 1.00i)33-s + 1.41·43-s − 49-s + (1.41 − 1.41i)51-s + (1.00 + 1.00i)57-s + 1.41i·59-s + 1.41·67-s + (−0.707 − 0.707i)75-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)3-s + 1.00i·9-s + 1.41i·11-s + 2i·17-s − 1.41·19-s + 25-s + (0.707 − 0.707i)27-s + (1.00 − 1.00i)33-s + 1.41·43-s − 49-s + (1.41 − 1.41i)51-s + (1.00 + 1.00i)57-s + 1.41i·59-s + 1.41·67-s + (−0.707 − 0.707i)75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7510711367\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7510711367\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.707 + 0.707i)T \) |
good | 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 - 1.41iT - T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 - 2iT - T^{2} \) |
| 19 | \( 1 + 1.41T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.41T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 1.41iT - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - 1.41T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - 1.41iT - T^{2} \) |
| 89 | \( 1 + 2iT - T^{2} \) |
| 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
−9.971116130565724134157014153831, −8.777771330674611926463490035241, −8.088559094822749314830387147590, −7.21641884535621143701813114205, −6.51550777352264895345966044221, −5.84504045857925779397450540147, −4.75180123181174105061516764535, −4.04833588597752672251228191017, −2.38832022748634222207639775930, −1.53603653617730186155043298715,
0.68119417147713322337564506440, 2.68248220446785032416315016012, 3.61448118192798378442395823001, 4.66686379699381178383873813382, 5.34020912141235155753531105717, 6.26068990860901136944665667563, 6.89078908662822911633680426017, 8.083274555376128056894866359870, 8.959691317762781180466982587569, 9.475815298205732827349826490842