L(s) = 1 | + (−0.707 + 0.707i)3-s + (0.707 + 0.707i)5-s + (−1 + i)7-s − 1.00i·9-s + 1.41·11-s − 1.00·15-s − 1.41i·21-s + 1.00i·25-s + (0.707 + 0.707i)27-s + 1.41·29-s + 2i·31-s + (−1.00 + 1.00i)33-s − 1.41·35-s + (0.707 − 0.707i)45-s − i·49-s + ⋯ |
L(s) = 1 | + (−0.707 + 0.707i)3-s + (0.707 + 0.707i)5-s + (−1 + i)7-s − 1.00i·9-s + 1.41·11-s − 1.00·15-s − 1.41i·21-s + 1.00i·25-s + (0.707 + 0.707i)27-s + 1.41·29-s + 2i·31-s + (−1.00 + 1.00i)33-s − 1.41·35-s + (0.707 − 0.707i)45-s − i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.049342516\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.049342516\) |
\(L(1)\) |
|
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 | \( 1 + (-0.707 - 0.707i)T \) |
good | 7 | \( 1 + (1 - i)T - iT^{2} \) |
| 11 | \( 1 - 1.41T + T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 - 1.41T + T^{2} \) |
| 31 | \( 1 - 2iT - T^{2} \) |
| 37 | \( 1 + iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + iT^{2} \) |
| 59 | \( 1 + 1.41iT - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1 - i)T - iT^{2} \) |
| 79 | \( 1 + 2T + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (1 + i)T + iT^{2} \) |
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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
−9.036355179053238494146492670375, −8.567028179050148826625836275312, −6.88423211119388348815977619253, −6.67611372517904895493517297027, −5.98296947007311282626318227127, −5.36666523351654791701013101223, −4.38930324200167017740298440733, −3.37260501868786465966310428448, −2.83834540598223457255251747146, −1.44962416317147434205457569449,
0.71726285832060724189576117864, 1.53845581928166252501954961494, 2.72916692512216691360611965048, 4.03064347551291914590175457524, 4.55564719869795247139301794331, 5.72115192245080830620559876807, 6.23878698680512864672170059301, 6.78888396317214201796209711148, 7.48690461028808060385452155045, 8.423657925685737771910314932554