| L(s) = 1 | + 4·2-s + 8·4-s − 5·5-s − 20·10-s − 32·11-s + 38·13-s − 64·16-s + 26·17-s − 100·19-s − 40·20-s − 128·22-s + 78·23-s + 25·25-s + 152·26-s + 50·29-s + 108·31-s − 256·32-s + 104·34-s + 266·37-s − 400·38-s + 22·41-s + 442·43-s − 256·44-s + 312·46-s − 514·47-s + 100·50-s + 304·52-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 0.447·5-s − 0.632·10-s − 0.877·11-s + 0.810·13-s − 16-s + 0.370·17-s − 1.20·19-s − 0.447·20-s − 1.24·22-s + 0.707·23-s + 1/5·25-s + 1.14·26-s + 0.320·29-s + 0.625·31-s − 1.41·32-s + 0.524·34-s + 1.18·37-s − 1.70·38-s + 0.0838·41-s + 1.56·43-s − 0.877·44-s + 1.00·46-s − 1.59·47-s + 0.282·50-s + 0.810·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.851807577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.851807577\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - p^{2} T + p^{3} T^{2} \) |
| 11 | \( 1 + 32 T + p^{3} T^{2} \) |
| 13 | \( 1 - 38 T + p^{3} T^{2} \) |
| 17 | \( 1 - 26 T + p^{3} T^{2} \) |
| 19 | \( 1 + 100 T + p^{3} T^{2} \) |
| 23 | \( 1 - 78 T + p^{3} T^{2} \) |
| 29 | \( 1 - 50 T + p^{3} T^{2} \) |
| 31 | \( 1 - 108 T + p^{3} T^{2} \) |
| 37 | \( 1 - 266 T + p^{3} T^{2} \) |
| 41 | \( 1 - 22 T + p^{3} T^{2} \) |
| 43 | \( 1 - 442 T + p^{3} T^{2} \) |
| 47 | \( 1 + 514 T + p^{3} T^{2} \) |
| 53 | \( 1 + 2 T + p^{3} T^{2} \) |
| 59 | \( 1 - 500 T + p^{3} T^{2} \) |
| 61 | \( 1 - 518 T + p^{3} T^{2} \) |
| 67 | \( 1 - 126 T + p^{3} T^{2} \) |
| 71 | \( 1 + 412 T + p^{3} T^{2} \) |
| 73 | \( 1 - 878 T + p^{3} T^{2} \) |
| 79 | \( 1 - 600 T + p^{3} T^{2} \) |
| 83 | \( 1 - 282 T + p^{3} T^{2} \) |
| 89 | \( 1 + 150 T + p^{3} T^{2} \) |
| 97 | \( 1 + 386 T + p^{3} T^{2} \) |
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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
−8.509486786941047839879870652584, −7.902208687944960329925029334043, −6.83560797154759942478092519872, −6.19280060816893817005010637999, −5.39498685320204620797063145562, −4.61916734316854745181635708417, −3.93508647247709037382339056980, −3.07799873870492823120162188917, −2.27031413535354370753077585488, −0.69551655067784332434603405593,
0.69551655067784332434603405593, 2.27031413535354370753077585488, 3.07799873870492823120162188917, 3.93508647247709037382339056980, 4.61916734316854745181635708417, 5.39498685320204620797063145562, 6.19280060816893817005010637999, 6.83560797154759942478092519872, 7.902208687944960329925029334043, 8.509486786941047839879870652584
