| L(s) = 1 | + 2·2-s − 4·3-s + 4·4-s − 8·6-s + 8·8-s − 11·9-s + 30·11-s − 16·12-s + 4·13-s + 16·16-s − 9·17-s − 22·18-s − 88·19-s + 60·22-s − 33·23-s − 32·24-s + 8·26-s + 152·27-s + 126·29-s + 155·31-s + 32·32-s − 120·33-s − 18·34-s − 44·36-s − 116·37-s − 176·38-s − 16·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.769·3-s + 1/2·4-s − 0.544·6-s + 0.353·8-s − 0.407·9-s + 0.822·11-s − 0.384·12-s + 0.0853·13-s + 1/4·16-s − 0.128·17-s − 0.288·18-s − 1.06·19-s + 0.581·22-s − 0.299·23-s − 0.272·24-s + 0.0603·26-s + 1.08·27-s + 0.806·29-s + 0.898·31-s + 0.176·32-s − 0.633·33-s − 0.0907·34-s − 0.203·36-s − 0.515·37-s − 0.751·38-s − 0.0656·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 30 T + p^{3} T^{2} \) |
| 13 | \( 1 - 4 T + p^{3} T^{2} \) |
| 17 | \( 1 + 9 T + p^{3} T^{2} \) |
| 19 | \( 1 + 88 T + p^{3} T^{2} \) |
| 23 | \( 1 + 33 T + p^{3} T^{2} \) |
| 29 | \( 1 - 126 T + p^{3} T^{2} \) |
| 31 | \( 1 - 5 p T + p^{3} T^{2} \) |
| 37 | \( 1 + 116 T + p^{3} T^{2} \) |
| 41 | \( 1 + 423 T + p^{3} T^{2} \) |
| 43 | \( 1 - 340 T + p^{3} T^{2} \) |
| 47 | \( 1 + 339 T + p^{3} T^{2} \) |
| 53 | \( 1 - 312 T + p^{3} T^{2} \) |
| 59 | \( 1 + 462 T + p^{3} T^{2} \) |
| 61 | \( 1 - 326 T + p^{3} T^{2} \) |
| 67 | \( 1 + 704 T + p^{3} T^{2} \) |
| 71 | \( 1 - 621 T + p^{3} T^{2} \) |
| 73 | \( 1 - 250 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1105 T + p^{3} T^{2} \) |
| 83 | \( 1 - 198 T + p^{3} T^{2} \) |
| 89 | \( 1 + 873 T + p^{3} T^{2} \) |
| 97 | \( 1 + 905 T + p^{3} T^{2} \) |
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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
−8.288826933649081804031307735945, −7.09895167971002988244359403603, −6.40858754516366698151847979247, −5.97385202001163081959031219435, −5.00129501667679256498438890173, −4.35989290328692548030608618733, −3.41389078278378289400261820596, −2.39714754807025098565274758171, −1.23862813108682627132707077882, 0,
1.23862813108682627132707077882, 2.39714754807025098565274758171, 3.41389078278378289400261820596, 4.35989290328692548030608618733, 5.00129501667679256498438890173, 5.97385202001163081959031219435, 6.40858754516366698151847979247, 7.09895167971002988244359403603, 8.288826933649081804031307735945
