L(s) = 1 | − 2-s + 4-s − 8-s − 2·11-s − 2·13-s + 16-s − 17-s + 4·19-s + 2·22-s + 2·23-s + 2·26-s − 2·29-s + 8·31-s − 32-s + 34-s − 6·37-s − 4·38-s − 10·41-s − 2·44-s − 2·46-s − 2·47-s − 2·52-s + 10·53-s + 2·58-s − 6·59-s + 10·61-s − 8·62-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 0.603·11-s − 0.554·13-s + 1/4·16-s − 0.242·17-s + 0.917·19-s + 0.426·22-s + 0.417·23-s + 0.392·26-s − 0.371·29-s + 1.43·31-s − 0.176·32-s + 0.171·34-s − 0.986·37-s − 0.648·38-s − 1.56·41-s − 0.301·44-s − 0.294·46-s − 0.291·47-s − 0.277·52-s + 1.37·53-s + 0.262·58-s − 0.781·59-s + 1.28·61-s − 1.01·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−12.62945648249673, −12.12198561382419, −11.80108587774253, −11.30513961668118, −10.92343429090258, −10.23731102060016, −10.02312588748701, −9.700047777788138, −9.041825209745646, −8.522478220508145, −8.337243783862367, −7.596285409978980, −7.367901509881167, −6.682670016303663, −6.560562388225635, −5.584557596432685, −5.378392975364338, −4.845272037572088, −4.246302315549603, −3.527296394663234, −3.057463640161345, −2.540295788486323, −1.992393100910320, −1.336379985124348, −0.6830046542997129, 0,
0.6830046542997129, 1.336379985124348, 1.992393100910320, 2.540295788486323, 3.057463640161345, 3.527296394663234, 4.246302315549603, 4.845272037572088, 5.378392975364338, 5.584557596432685, 6.560562388225635, 6.682670016303663, 7.367901509881167, 7.596285409978980, 8.337243783862367, 8.522478220508145, 9.041825209745646, 9.700047777788138, 10.02312588748701, 10.23731102060016, 10.92343429090258, 11.30513961668118, 11.80108587774253, 12.12198561382419, 12.62945648249673