| L(s) = 1 | − 3.79i·5-s + 2·7-s + 3.79·11-s − 0.791i·13-s − 1.58i·17-s − 7.58i·19-s − 0.791i·23-s − 9.37·25-s − 0.791i·29-s + 5.37i·31-s − 7.58i·35-s + (4 − 4.58i)37-s − 5.20·41-s − 6i·43-s + 1.58·47-s + ⋯ |
| L(s) = 1 | − 1.69i·5-s + 0.755·7-s + 1.14·11-s − 0.219i·13-s − 0.383i·17-s − 1.73i·19-s − 0.164i·23-s − 1.87·25-s − 0.146i·29-s + 0.965i·31-s − 1.28i·35-s + (0.657 − 0.753i)37-s − 0.813·41-s − 0.914i·43-s + 0.230·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.657 + 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.107212112\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.107212112\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 + (-4 + 4.58i)T \) |
| good | 5 | \( 1 + 3.79iT - 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 3.79T + 11T^{2} \) |
| 13 | \( 1 + 0.791iT - 13T^{2} \) |
| 17 | \( 1 + 1.58iT - 17T^{2} \) |
| 19 | \( 1 + 7.58iT - 19T^{2} \) |
| 23 | \( 1 + 0.791iT - 23T^{2} \) |
| 29 | \( 1 + 0.791iT - 29T^{2} \) |
| 31 | \( 1 - 5.37iT - 31T^{2} \) |
| 41 | \( 1 + 5.20T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 - 1.58T + 47T^{2} \) |
| 53 | \( 1 + 7.58T + 53T^{2} \) |
| 59 | \( 1 - 7.58iT - 59T^{2} \) |
| 61 | \( 1 + 8.20iT - 61T^{2} \) |
| 67 | \( 1 - 7.37T + 67T^{2} \) |
| 71 | \( 1 - 9.16T + 71T^{2} \) |
| 73 | \( 1 - 9.37T + 73T^{2} \) |
| 79 | \( 1 - 12.7iT - 79T^{2} \) |
| 83 | \( 1 + 3.16T + 83T^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 - 4.41iT - 97T^{2} \) |
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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
−8.240086595617072990718725431644, −7.21668597186653114202398472202, −6.55366336115781686471457905942, −5.49344409748862643523422614567, −4.93082653231151317846145609150, −4.46233761736697343050332044544, −3.58872422416041769871482616192, −2.28446657628213135367123426933, −1.32435715612327475855641149183, −0.58322474450299702513831097077,
1.45114693371317707913388312876, 2.17404410218911141856966905259, 3.30909649504547511072208441764, 3.79858338191169812623221172671, 4.67744365270094587679961482686, 5.85805979678448905595658823666, 6.33904941097414211940476343545, 6.91157424251483425544744415932, 7.85352532966755362539248143910, 8.106695834817074279043648424304
