L(s) = 1 | − 3-s − 5-s + 7-s + 9-s + 2·13-s + 15-s + 2·17-s − 21-s + 25-s − 27-s + 2·29-s − 4·31-s − 35-s + 6·37-s − 2·39-s − 6·41-s − 12·43-s − 45-s + 4·47-s + 49-s − 2·51-s + 6·53-s − 4·59-s + 10·61-s + 63-s − 2·65-s − 4·67-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.554·13-s + 0.258·15-s + 0.485·17-s − 0.218·21-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.718·31-s − 0.169·35-s + 0.986·37-s − 0.320·39-s − 0.937·41-s − 1.82·43-s − 0.149·45-s + 0.583·47-s + 1/7·49-s − 0.280·51-s + 0.824·53-s − 0.520·59-s + 1.28·61-s + 0.125·63-s − 0.248·65-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.80505137111683, −13.56284721718321, −12.96010015008527, −12.39972786099849, −12.00834811913115, −11.50831312542739, −11.12696469386194, −10.65960100226546, −10.06134154213383, −9.707633795922635, −8.900054710414234, −8.535248105054871, −7.917196668864914, −7.550286719178617, −6.824382449242051, −6.474815954249291, −5.824642248390229, −5.166506228051865, −4.942538815751146, −4.031293705492811, −3.748666153469534, −3.013861351171279, −2.233919656412577, −1.468588077939270, −0.8744093128993339, 0,
0.8744093128993339, 1.468588077939270, 2.233919656412577, 3.013861351171279, 3.748666153469534, 4.031293705492811, 4.942538815751146, 5.166506228051865, 5.824642248390229, 6.474815954249291, 6.824382449242051, 7.550286719178617, 7.917196668864914, 8.535248105054871, 8.900054710414234, 9.707633795922635, 10.06134154213383, 10.65960100226546, 11.12696469386194, 11.50831312542739, 12.00834811913115, 12.39972786099849, 12.96010015008527, 13.56284721718321, 13.80505137111683