L(s) = 1 | + (−1.30 + 1.30i)2-s − 2.41i·4-s + (0.923 + 0.382i)5-s + (1.84 + 1.84i)8-s + (−1.70 + 0.707i)10-s − 1.84i·11-s + (0.707 − 0.707i)13-s − 2.41·16-s + (0.923 − 2.23i)20-s + (2.41 + 2.41i)22-s + (0.707 + 0.707i)25-s + 1.84i·26-s + (1.30 − 1.30i)32-s + (1.00 + 2.41i)40-s + 0.765i·41-s + ⋯ |
L(s) = 1 | + (−1.30 + 1.30i)2-s − 2.41i·4-s + (0.923 + 0.382i)5-s + (1.84 + 1.84i)8-s + (−1.70 + 0.707i)10-s − 1.84i·11-s + (0.707 − 0.707i)13-s − 2.41·16-s + (0.923 − 2.23i)20-s + (2.41 + 2.41i)22-s + (0.707 + 0.707i)25-s + 1.84i·26-s + (1.30 − 1.30i)32-s + (1.00 + 2.41i)40-s + 0.765i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5680328029\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5680328029\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.923 - 0.382i)T \) |
| 13 | \( 1 + (-0.707 + 0.707i)T \) |
good | 2 | \( 1 + (1.30 - 1.30i)T - iT^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + 1.84iT - T^{2} \) |
| 17 | \( 1 + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 - 0.765iT - T^{2} \) |
| 43 | \( 1 + (1 - i)T - iT^{2} \) |
| 47 | \( 1 + (0.541 - 0.541i)T - iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - 0.765T + T^{2} \) |
| 61 | \( 1 + 1.41T + T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + 0.765iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - 1.41iT - T^{2} \) |
| 83 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
| 89 | \( 1 - 1.84T + T^{2} \) |
| 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
−10.73860052340098912597415889779, −9.946169014876179928628885276373, −9.087160265251616816100968054184, −8.411154622757731180766813334356, −7.67026016905902067042344895413, −6.35638345925590981295990084040, −6.10427129475741550000436138089, −5.18515876633768035895774300738, −3.10246286632800983312555546459, −1.23527477376490119975142196625,
1.59874041401367885536930546194, 2.27312621645193045876677843939, 3.80276706457507418806013034548, 4.95187547651253930837016627383, 6.59483893430629035520582645665, 7.49277186874296420089037597519, 8.655214474780048353855139455041, 9.189510731008228967679438582306, 10.02986158880638050593872769127, 10.39916256938488087519610781854