| L(s) = 1 | − 2.41·2-s + 3.84·4-s − 4.01·5-s + 7-s − 4.45·8-s + 9.69·10-s − 0.690·13-s − 2.41·14-s + 3.08·16-s − 5.78·17-s − 1.80·19-s − 15.4·20-s − 2.03·23-s + 11.0·25-s + 1.66·26-s + 3.84·28-s − 7.98·29-s − 10.4·31-s + 1.46·32-s + 13.9·34-s − 4.01·35-s − 7.44·37-s + 4.35·38-s + 17.8·40-s + 5.66·41-s + 3.44·43-s + 4.92·46-s + ⋯ |
| L(s) = 1 | − 1.70·2-s + 1.92·4-s − 1.79·5-s + 0.377·7-s − 1.57·8-s + 3.06·10-s − 0.191·13-s − 0.646·14-s + 0.770·16-s − 1.40·17-s − 0.413·19-s − 3.44·20-s − 0.424·23-s + 2.21·25-s + 0.327·26-s + 0.726·28-s − 1.48·29-s − 1.88·31-s + 0.258·32-s + 2.39·34-s − 0.677·35-s − 1.22·37-s + 0.706·38-s + 2.82·40-s + 0.884·41-s + 0.524·43-s + 0.725·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04197968312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04197968312\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 5 | \( 1 + 4.01T + 5T^{2} \) |
| 13 | \( 1 + 0.690T + 13T^{2} \) |
| 17 | \( 1 + 5.78T + 17T^{2} \) |
| 19 | \( 1 + 1.80T + 19T^{2} \) |
| 23 | \( 1 + 2.03T + 23T^{2} \) |
| 29 | \( 1 + 7.98T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 + 7.44T + 37T^{2} \) |
| 41 | \( 1 - 5.66T + 41T^{2} \) |
| 43 | \( 1 - 3.44T + 43T^{2} \) |
| 47 | \( 1 + 7.32T + 47T^{2} \) |
| 53 | \( 1 - 1.79T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 + 6.95T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 - 7.20T + 73T^{2} \) |
| 79 | \( 1 + 5.33T + 79T^{2} \) |
| 83 | \( 1 - 8.59T + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 + 9.50T + 97T^{2} \) |
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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
−7.87828038224014164708853105330, −7.43784509989996977401043348057, −7.01506260132522249134776003715, −6.15110677027251436506837182302, −4.97049829643936005914950219828, −4.16864483190678098473361809532, −3.48933596935442727340487824716, −2.34881555771840583367376309363, −1.53290341171665545339208318797, −0.13277571471093191317180458763,
0.13277571471093191317180458763, 1.53290341171665545339208318797, 2.34881555771840583367376309363, 3.48933596935442727340487824716, 4.16864483190678098473361809532, 4.97049829643936005914950219828, 6.15110677027251436506837182302, 7.01506260132522249134776003715, 7.43784509989996977401043348057, 7.87828038224014164708853105330
