| L(s) = 1 | + 4·2-s + 8·4-s + 21·7-s − 11·11-s − 68·13-s + 84·14-s − 64·16-s − 21·17-s + 125·19-s − 44·22-s − 137·23-s − 272·26-s + 168·28-s + 150·29-s + 292·31-s − 256·32-s − 84·34-s − 349·37-s + 500·38-s − 497·41-s − 208·43-s − 88·44-s − 548·46-s + 369·47-s + 98·49-s − 544·52-s − 542·53-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.13·7-s − 0.301·11-s − 1.45·13-s + 1.60·14-s − 16-s − 0.299·17-s + 1.50·19-s − 0.426·22-s − 1.24·23-s − 2.05·26-s + 1.13·28-s + 0.960·29-s + 1.69·31-s − 1.41·32-s − 0.423·34-s − 1.55·37-s + 2.13·38-s − 1.89·41-s − 0.737·43-s − 0.301·44-s − 1.75·46-s + 1.14·47-s + 2/7·49-s − 1.45·52-s − 1.40·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + p T \) |
| good | 2 | \( 1 - p^{2} T + p^{3} T^{2} \) |
| 7 | \( 1 - 3 p T + p^{3} T^{2} \) |
| 13 | \( 1 + 68 T + p^{3} T^{2} \) |
| 17 | \( 1 + 21 T + p^{3} T^{2} \) |
| 19 | \( 1 - 125 T + p^{3} T^{2} \) |
| 23 | \( 1 + 137 T + p^{3} T^{2} \) |
| 29 | \( 1 - 150 T + p^{3} T^{2} \) |
| 31 | \( 1 - 292 T + p^{3} T^{2} \) |
| 37 | \( 1 + 349 T + p^{3} T^{2} \) |
| 41 | \( 1 + 497 T + p^{3} T^{2} \) |
| 43 | \( 1 + 208 T + p^{3} T^{2} \) |
| 47 | \( 1 - 369 T + p^{3} T^{2} \) |
| 53 | \( 1 + 542 T + p^{3} T^{2} \) |
| 59 | \( 1 + 235 T + p^{3} T^{2} \) |
| 61 | \( 1 - 482 T + p^{3} T^{2} \) |
| 67 | \( 1 + 734 T + p^{3} T^{2} \) |
| 71 | \( 1 + 587 T + p^{3} T^{2} \) |
| 73 | \( 1 + 518 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1045 T + p^{3} T^{2} \) |
| 83 | \( 1 - 608 T + p^{3} T^{2} \) |
| 89 | \( 1 - 770 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1541 T + p^{3} T^{2} \) |
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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
−8.031042035894685574087543035001, −7.30124261258908812301249042683, −6.49089983900888848649594555924, −5.50933261533434182196862861747, −4.87832284082544196129598447058, −4.53385816876557817827617212361, −3.34589146085272231575421692802, −2.57400490111776663057813107876, −1.59398473001742544126983924521, 0,
1.59398473001742544126983924521, 2.57400490111776663057813107876, 3.34589146085272231575421692802, 4.53385816876557817827617212361, 4.87832284082544196129598447058, 5.50933261533434182196862861747, 6.49089983900888848649594555924, 7.30124261258908812301249042683, 8.031042035894685574087543035001
