| L(s) = 1 | − 1.79·2-s + 1.20·4-s + 1.41·8-s − 3·9-s − 6.58·11-s − 4.95·16-s + 5.37·18-s + 11.7·22-s + 8.58·23-s + 8.16·29-s + 6.04·32-s − 3.62·36-s − 12.1·37-s − 1.41·43-s − 7.95·44-s − 15.3·46-s + 10·53-s − 14.6·58-s − 0.912·64-s + 15.7·67-s − 3.41·71-s − 4.25·72-s + 21.7·74-s + 9.74·79-s + 9·81-s + 2.53·86-s − 9.33·88-s + ⋯ |
| L(s) = 1 | − 1.26·2-s + 0.604·4-s + 0.501·8-s − 9-s − 1.98·11-s − 1.23·16-s + 1.26·18-s + 2.51·22-s + 1.78·23-s + 1.51·29-s + 1.06·32-s − 0.604·36-s − 1.99·37-s − 0.216·43-s − 1.19·44-s − 2.26·46-s + 1.37·53-s − 1.92·58-s − 0.114·64-s + 1.92·67-s − 0.405·71-s − 0.501·72-s + 2.53·74-s + 1.09·79-s + 81-s + 0.273·86-s − 0.994·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5248061917\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5248061917\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.79T + 2T^{2} \) |
| 3 | \( 1 + 3T^{2} \) |
| 11 | \( 1 + 6.58T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 8.58T + 23T^{2} \) |
| 29 | \( 1 - 8.16T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 12.1T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 1.41T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 15.7T + 67T^{2} \) |
| 71 | \( 1 + 3.41T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 9.74T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−9.699600750910145630848530593743, −8.641272752858889519330169278661, −8.428496916064441131782653601895, −7.50444860976862356443729336925, −6.74953425220276383926801424219, −5.39576834090927669355728915322, −4.86967648718154815102593211128, −3.16583587986373159734214660538, −2.28095635488370275222297630382, −0.62722834978033193815539962648,
0.62722834978033193815539962648, 2.28095635488370275222297630382, 3.16583587986373159734214660538, 4.86967648718154815102593211128, 5.39576834090927669355728915322, 6.74953425220276383926801424219, 7.50444860976862356443729336925, 8.428496916064441131782653601895, 8.641272752858889519330169278661, 9.699600750910145630848530593743
