L(s) = 1 | + 2i·5-s − 2·7-s + 3·9-s + 2i·11-s + i·13-s + 6·17-s − 6i·19-s + 8·23-s + 25-s − 2i·29-s − 10·31-s − 4i·35-s − 6i·37-s + 6·41-s − 4i·43-s + ⋯ |
L(s) = 1 | + 0.894i·5-s − 0.755·7-s + 9-s + 0.603i·11-s + 0.277i·13-s + 1.45·17-s − 1.37i·19-s + 1.66·23-s + 0.200·25-s − 0.371i·29-s − 1.79·31-s − 0.676i·35-s − 0.986i·37-s + 0.937·41-s − 0.609i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.981535823\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.981535823\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 - iT \) |
good | 3 | \( 1 - 3T^{2} \) |
| 5 | \( 1 - 2iT - 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 10iT - 59T^{2} \) |
| 61 | \( 1 - 2iT - 61T^{2} \) |
| 67 | \( 1 - 10iT - 67T^{2} \) |
| 71 | \( 1 - 10T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.090322850212685434003947732148, −7.57280546991219335579010534867, −7.19473966854861786710649131782, −6.75804817318721114003935994198, −5.74790785468569100893343031948, −4.89100168812448932038231351354, −3.92811730028493936210430088391, −3.14981636149932821211840835164, −2.32121538376297524152832952161, −0.978048321201144394584188386267,
0.810237419234173064450958648354, 1.61858460732684329621560685007, 3.22506366195855257825503171715, 3.64429077591181687790501944794, 4.83534112165499395539914685161, 5.41428232339832053708538941923, 6.24213511517197549276612964489, 7.10041375439695171027985830439, 7.84246254566311702676034805965, 8.501790641516419792698386485765