L(s) = 1 | + 2-s + 2.44·3-s + 4-s + 2.44·6-s − 2.44·7-s + 8-s + 2.99·9-s − 11-s + 2.44·12-s − 3.89·13-s − 2.44·14-s + 16-s − 4.44·17-s + 2.99·18-s − 5.99·21-s − 22-s + 0.898·23-s + 2.44·24-s − 5·25-s − 3.89·26-s − 2.44·28-s + 1.89·29-s − 2·31-s + 32-s − 2.44·33-s − 4.44·34-s + 2.99·36-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.41·3-s + 0.5·4-s + 0.999·6-s − 0.925·7-s + 0.353·8-s + 0.999·9-s − 0.301·11-s + 0.707·12-s − 1.08·13-s − 0.654·14-s + 0.250·16-s − 1.07·17-s + 0.707·18-s − 1.30·21-s − 0.213·22-s + 0.187·23-s + 0.499·24-s − 25-s − 0.764·26-s − 0.462·28-s + 0.352·29-s − 0.359·31-s + 0.176·32-s − 0.426·33-s − 0.763·34-s + 0.499·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 - 2.44T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 + 2.44T + 7T^{2} \) |
| 13 | \( 1 + 3.89T + 13T^{2} \) |
| 17 | \( 1 + 4.44T + 17T^{2} \) |
| 23 | \( 1 - 0.898T + 23T^{2} \) |
| 29 | \( 1 - 1.89T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 8.44T + 37T^{2} \) |
| 41 | \( 1 - 7.55T + 41T^{2} \) |
| 43 | \( 1 - 9.44T + 43T^{2} \) |
| 47 | \( 1 + 6.55T + 47T^{2} \) |
| 53 | \( 1 - 7.79T + 53T^{2} \) |
| 59 | \( 1 + 2.89T + 59T^{2} \) |
| 61 | \( 1 - 7.89T + 61T^{2} \) |
| 67 | \( 1 + 9.55T + 67T^{2} \) |
| 71 | \( 1 + 7.44T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 + 4.44T + 79T^{2} \) |
| 83 | \( 1 + 5.44T + 83T^{2} \) |
| 89 | \( 1 - 9T + 89T^{2} \) |
| 97 | \( 1 + 3.89T + 97T^{2} \) |
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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
−7.41104457647287623196978300601, −6.96989266795564616399652650539, −6.11587500044652159904801716096, −5.35486028256722936225129461557, −4.38930227901307410041048054809, −3.87483994020281898592960821314, −2.95927379618451889654316078549, −2.58509889236569663670868120807, −1.78622592631007793023103371042, 0,
1.78622592631007793023103371042, 2.58509889236569663670868120807, 2.95927379618451889654316078549, 3.87483994020281898592960821314, 4.38930227901307410041048054809, 5.35486028256722936225129461557, 6.11587500044652159904801716096, 6.96989266795564616399652650539, 7.41104457647287623196978300601