| L(s) = 1 | − 2.54·5-s − 7-s + 5.35·11-s + 6.80·13-s + 4.03·17-s − 19-s − 7.79·23-s + 1.45·25-s + 2.81·29-s − 1.54·31-s + 2.54·35-s + 5.09·37-s − 9.22·41-s + 1.54·43-s − 5.46·47-s + 49-s + 9.39·53-s − 13.6·55-s + 5.09·61-s − 17.2·65-s − 3.25·67-s + 1.32·71-s + 16.5·73-s − 5.35·77-s + 11.2·79-s − 6.19·83-s − 10.2·85-s + ⋯ |
| L(s) = 1 | − 1.13·5-s − 0.377·7-s + 1.61·11-s + 1.88·13-s + 0.978·17-s − 0.229·19-s − 1.62·23-s + 0.290·25-s + 0.523·29-s − 0.277·31-s + 0.429·35-s + 0.837·37-s − 1.44·41-s + 0.235·43-s − 0.797·47-s + 0.142·49-s + 1.28·53-s − 1.83·55-s + 0.651·61-s − 2.14·65-s − 0.398·67-s + 0.157·71-s + 1.93·73-s − 0.610·77-s + 1.26·79-s − 0.679·83-s − 1.11·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.731095414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.731095414\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 2.54T + 5T^{2} \) |
| 11 | \( 1 - 5.35T + 11T^{2} \) |
| 13 | \( 1 - 6.80T + 13T^{2} \) |
| 17 | \( 1 - 4.03T + 17T^{2} \) |
| 23 | \( 1 + 7.79T + 23T^{2} \) |
| 29 | \( 1 - 2.81T + 29T^{2} \) |
| 31 | \( 1 + 1.54T + 31T^{2} \) |
| 37 | \( 1 - 5.09T + 37T^{2} \) |
| 41 | \( 1 + 9.22T + 41T^{2} \) |
| 43 | \( 1 - 1.54T + 43T^{2} \) |
| 47 | \( 1 + 5.46T + 47T^{2} \) |
| 53 | \( 1 - 9.39T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 5.09T + 61T^{2} \) |
| 67 | \( 1 + 3.25T + 67T^{2} \) |
| 71 | \( 1 - 1.32T + 71T^{2} \) |
| 73 | \( 1 - 16.5T + 73T^{2} \) |
| 79 | \( 1 - 11.2T + 79T^{2} \) |
| 83 | \( 1 + 6.19T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 + 17.9T + 97T^{2} \) |
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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
−8.299188409379086985163201407174, −7.70978490289662335078219414827, −6.66309533391090313085671419640, −6.31409246288072125601485836429, −5.46708088910072345842167089718, −4.08050975262303829175866257678, −3.91820568119709272520041352285, −3.23757888162928826721489264790, −1.71681658209574510326074413655, −0.76654623292882484452203057510,
0.76654623292882484452203057510, 1.71681658209574510326074413655, 3.23757888162928826721489264790, 3.91820568119709272520041352285, 4.08050975262303829175866257678, 5.46708088910072345842167089718, 6.31409246288072125601485836429, 6.66309533391090313085671419640, 7.70978490289662335078219414827, 8.299188409379086985163201407174
