| L(s) = 1 | − 2-s + 4-s − 8-s − 4·11-s − 13-s + 16-s + 6·17-s + 4·22-s + 2·23-s + 26-s − 6·29-s + 2·31-s − 32-s − 6·34-s + 6·37-s + 2·41-s + 8·43-s − 4·44-s − 2·46-s − 8·47-s − 52-s + 6·58-s + 4·59-s − 10·61-s − 2·62-s + 64-s − 2·67-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.20·11-s − 0.277·13-s + 1/4·16-s + 1.45·17-s + 0.852·22-s + 0.417·23-s + 0.196·26-s − 1.11·29-s + 0.359·31-s − 0.176·32-s − 1.02·34-s + 0.986·37-s + 0.312·41-s + 1.21·43-s − 0.603·44-s − 0.294·46-s − 1.16·47-s − 0.138·52-s + 0.787·58-s + 0.520·59-s − 1.28·61-s − 0.254·62-s + 1/8·64-s − 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.257117818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.257117818\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−12.71150024324449, −12.18413185283187, −11.83523969572989, −11.18028519806410, −10.86359184212739, −10.41594617532627, −9.834329871809782, −9.672708591369522, −9.047844477421259, −8.581120639079730, −7.887782522302603, −7.704709914846228, −7.392630855002846, −6.713496276900358, −6.064018571260457, −5.678631648332718, −5.228681118461510, −4.650935661914770, −4.025200574547895, −3.273512007738642, −2.907777018073037, −2.367676357038459, −1.684352276078979, −1.037150748059146, −0.3749066368411541,
0.3749066368411541, 1.037150748059146, 1.684352276078979, 2.367676357038459, 2.907777018073037, 3.273512007738642, 4.025200574547895, 4.650935661914770, 5.228681118461510, 5.678631648332718, 6.064018571260457, 6.713496276900358, 7.392630855002846, 7.704709914846228, 7.887782522302603, 8.581120639079730, 9.047844477421259, 9.672708591369522, 9.834329871809782, 10.41594617532627, 10.86359184212739, 11.18028519806410, 11.83523969572989, 12.18413185283187, 12.71150024324449
