| L(s) = 1 | + 11·5-s + 30·7-s − 4·11-s + 16·13-s + 70·17-s + 5·19-s + 53·23-s − 4·25-s − 15·29-s + 74·31-s + 330·35-s − 156·37-s + 366·41-s + 292·43-s − 201·47-s + 557·49-s + 557·53-s − 44·55-s − 364·59-s − 518·61-s + 176·65-s − 7·67-s − 473·71-s − 359·73-s − 120·77-s + 1.02e3·79-s + 1.07e3·83-s + ⋯ |
| L(s) = 1 | + 0.983·5-s + 1.61·7-s − 0.109·11-s + 0.341·13-s + 0.998·17-s + 0.0603·19-s + 0.480·23-s − 0.0319·25-s − 0.0960·29-s + 0.428·31-s + 1.59·35-s − 0.693·37-s + 1.39·41-s + 1.03·43-s − 0.623·47-s + 1.62·49-s + 1.44·53-s − 0.107·55-s − 0.803·59-s − 1.08·61-s + 0.335·65-s − 0.0127·67-s − 0.790·71-s − 0.575·73-s − 0.177·77-s + 1.45·79-s + 1.42·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.894388046\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.894388046\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 11 T + p^{3} T^{2} \) |
| 7 | \( 1 - 30 T + p^{3} T^{2} \) |
| 11 | \( 1 + 4 T + p^{3} T^{2} \) |
| 13 | \( 1 - 16 T + p^{3} T^{2} \) |
| 17 | \( 1 - 70 T + p^{3} T^{2} \) |
| 19 | \( 1 - 5 T + p^{3} T^{2} \) |
| 23 | \( 1 - 53 T + p^{3} T^{2} \) |
| 29 | \( 1 + 15 T + p^{3} T^{2} \) |
| 31 | \( 1 - 74 T + p^{3} T^{2} \) |
| 37 | \( 1 + 156 T + p^{3} T^{2} \) |
| 41 | \( 1 - 366 T + p^{3} T^{2} \) |
| 43 | \( 1 - 292 T + p^{3} T^{2} \) |
| 47 | \( 1 + 201 T + p^{3} T^{2} \) |
| 53 | \( 1 - 557 T + p^{3} T^{2} \) |
| 59 | \( 1 + 364 T + p^{3} T^{2} \) |
| 61 | \( 1 + 518 T + p^{3} T^{2} \) |
| 67 | \( 1 + 7 T + p^{3} T^{2} \) |
| 71 | \( 1 + 473 T + p^{3} T^{2} \) |
| 73 | \( 1 + 359 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1024 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1078 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1044 T + p^{3} T^{2} \) |
| 97 | \( 1 + 193 T + p^{3} T^{2} \) |
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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
−8.880428275476718620047126575770, −7.966413157218690360521659244885, −7.48344808062876336911505965538, −6.29718587552492723877491267117, −5.54129946261652372079275698088, −4.95829258166693975933336631802, −3.97808863694596373715175358254, −2.69857054126434169801200962863, −1.75043023755799458745119324405, −1.00421811453066494265703353986,
1.00421811453066494265703353986, 1.75043023755799458745119324405, 2.69857054126434169801200962863, 3.97808863694596373715175358254, 4.95829258166693975933336631802, 5.54129946261652372079275698088, 6.29718587552492723877491267117, 7.48344808062876336911505965538, 7.966413157218690360521659244885, 8.880428275476718620047126575770
