| L(s) = 1 | + 3·4-s + 2·5-s + 5·9-s + 5·16-s − 12·19-s + 6·20-s − 25-s − 14·29-s + 4·31-s + 15·36-s + 10·41-s + 10·45-s − 20·59-s + 14·61-s + 3·64-s − 4·71-s − 36·76-s + 4·79-s + 10·80-s + 16·81-s − 18·89-s − 24·95-s − 3·100-s + 18·101-s − 10·109-s − 42·116-s − 22·121-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 0.894·5-s + 5/3·9-s + 5/4·16-s − 2.75·19-s + 1.34·20-s − 1/5·25-s − 2.59·29-s + 0.718·31-s + 5/2·36-s + 1.56·41-s + 1.49·45-s − 2.60·59-s + 1.79·61-s + 3/8·64-s − 0.474·71-s − 4.12·76-s + 0.450·79-s + 1.11·80-s + 16/9·81-s − 1.90·89-s − 2.46·95-s − 0.299·100-s + 1.79·101-s − 0.957·109-s − 3.89·116-s − 2·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.443015397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.443015397\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.46951940122934231732586921988, −11.84681627409977531747499048886, −11.13252314025887071708034118484, −10.98756286558056088332578216807, −10.32375677978690018736264978442, −10.25327295647993626477769852673, −9.351854607884083566221417858624, −9.271730083406293209770252837771, −8.315789301112807317702033567314, −7.73218485914141313641570698096, −7.29763084938530033112741735749, −6.80122903032974038134729394574, −6.21699066979898812751951526873, −6.06457335607048116366724981930, −5.22034242659032869267184379667, −4.22970151780495777256668162325, −3.99128029617419054937583513285, −2.75196144789082061150707603900, −1.96757377625627144750002423327, −1.71885448512894583614551319232,
1.71885448512894583614551319232, 1.96757377625627144750002423327, 2.75196144789082061150707603900, 3.99128029617419054937583513285, 4.22970151780495777256668162325, 5.22034242659032869267184379667, 6.06457335607048116366724981930, 6.21699066979898812751951526873, 6.80122903032974038134729394574, 7.29763084938530033112741735749, 7.73218485914141313641570698096, 8.315789301112807317702033567314, 9.271730083406293209770252837771, 9.351854607884083566221417858624, 10.25327295647993626477769852673, 10.32375677978690018736264978442, 10.98756286558056088332578216807, 11.13252314025887071708034118484, 11.84681627409977531747499048886, 12.46951940122934231732586921988
