L(s) = 1 | + 2·2-s + 2·4-s − 5-s − 2·7-s − 2·10-s − 3·11-s − 4·13-s − 4·14-s − 4·16-s − 2·17-s − 2·19-s − 2·20-s − 6·22-s + 5·23-s + 25-s − 8·26-s − 4·28-s − 29-s + 2·31-s − 8·32-s − 4·34-s + 2·35-s − 5·37-s − 4·38-s − 41-s − 43-s − 6·44-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 4-s − 0.447·5-s − 0.755·7-s − 0.632·10-s − 0.904·11-s − 1.10·13-s − 1.06·14-s − 16-s − 0.485·17-s − 0.458·19-s − 0.447·20-s − 1.27·22-s + 1.04·23-s + 1/5·25-s − 1.56·26-s − 0.755·28-s − 0.185·29-s + 0.359·31-s − 1.41·32-s − 0.685·34-s + 0.338·35-s − 0.821·37-s − 0.648·38-s − 0.156·41-s − 0.152·43-s − 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 + T \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 7 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−9.274013153839542218078154494445, −8.376552658149478662626812787318, −7.21472907123974619082067797266, −6.71645242034608803138890556636, −5.62356802838365750040538035218, −4.94164430757872187509488653621, −4.13302570378768496109961966402, −3.12099476831209731481380332558, −2.42429129958115379819484349060, 0,
2.42429129958115379819484349060, 3.12099476831209731481380332558, 4.13302570378768496109961966402, 4.94164430757872187509488653621, 5.62356802838365750040538035218, 6.71645242034608803138890556636, 7.21472907123974619082067797266, 8.376552658149478662626812787318, 9.274013153839542218078154494445