| L(s) = 1 | − 3-s − 5-s + 2·7-s + 9-s − 2·13-s + 15-s + 2·19-s − 2·21-s + 25-s − 27-s − 8·31-s − 2·35-s + 2·37-s + 2·39-s + 2·43-s − 45-s − 3·49-s + 6·53-s − 2·57-s + 12·59-s − 2·61-s + 2·63-s + 2·65-s + 4·67-s − 2·73-s − 75-s − 10·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.554·13-s + 0.258·15-s + 0.458·19-s − 0.436·21-s + 1/5·25-s − 0.192·27-s − 1.43·31-s − 0.338·35-s + 0.328·37-s + 0.320·39-s + 0.304·43-s − 0.149·45-s − 3/7·49-s + 0.824·53-s − 0.264·57-s + 1.56·59-s − 0.256·61-s + 0.251·63-s + 0.248·65-s + 0.488·67-s − 0.234·73-s − 0.115·75-s − 1.12·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−15.44483502893758, −14.89124582652719, −14.42907766044430, −14.02167760509917, −13.09142596067046, −12.83198980827481, −12.13546628766300, −11.59588660682130, −11.30819874944633, −10.72575475192156, −10.09253966709371, −9.588358227213143, −8.838014698962115, −8.349462702606710, −7.565787209454407, −7.302492546845624, −6.649173002182464, −5.780592188577633, −5.350801022960777, −4.740594928389498, −4.138895057750080, −3.482214426431978, −2.577267498936715, −1.795242248818977, −0.9695084419179220, 0,
0.9695084419179220, 1.795242248818977, 2.577267498936715, 3.482214426431978, 4.138895057750080, 4.740594928389498, 5.350801022960777, 5.780592188577633, 6.649173002182464, 7.302492546845624, 7.565787209454407, 8.349462702606710, 8.838014698962115, 9.588358227213143, 10.09253966709371, 10.72575475192156, 11.30819874944633, 11.59588660682130, 12.13546628766300, 12.83198980827481, 13.09142596067046, 14.02167760509917, 14.42907766044430, 14.89124582652719, 15.44483502893758
