| L(s) = 1 | + 2·3-s − 2·4-s − 5-s − 4·7-s + 9-s + 3·11-s − 4·12-s − 2·13-s − 2·15-s + 4·16-s + 6·17-s + 2·20-s − 8·21-s + 25-s − 4·27-s + 8·28-s + 3·29-s + 7·31-s + 6·33-s + 4·35-s − 2·36-s − 8·37-s − 4·39-s + 6·41-s − 4·43-s − 6·44-s − 45-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 4-s − 0.447·5-s − 1.51·7-s + 1/3·9-s + 0.904·11-s − 1.15·12-s − 0.554·13-s − 0.516·15-s + 16-s + 1.45·17-s + 0.447·20-s − 1.74·21-s + 1/5·25-s − 0.769·27-s + 1.51·28-s + 0.557·29-s + 1.25·31-s + 1.04·33-s + 0.676·35-s − 1/3·36-s − 1.31·37-s − 0.640·39-s + 0.937·41-s − 0.609·43-s − 0.904·44-s − 0.149·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.504775435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.504775435\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−9.180269078376159410057067247792, −8.641046451413056059928979018312, −7.888671076690515591144389934644, −7.06260344577378957035420993766, −6.11223643147922350650706661543, −5.09286214690838306892529783179, −3.83978926023296414656361190013, −3.53808712262848033914246425072, −2.62075476335234995502737434357, −0.795184987854384961401443682104,
0.795184987854384961401443682104, 2.62075476335234995502737434357, 3.53808712262848033914246425072, 3.83978926023296414656361190013, 5.09286214690838306892529783179, 6.11223643147922350650706661543, 7.06260344577378957035420993766, 7.888671076690515591144389934644, 8.641046451413056059928979018312, 9.180269078376159410057067247792
