L(s) = 1 | + (0.690 − 0.951i)3-s − 4-s + (0.809 − 0.587i)5-s + (−0.118 − 0.363i)9-s + (−0.690 + 0.951i)12-s − 1.17i·15-s + 16-s + (−0.809 + 0.587i)20-s + (−1.11 − 1.53i)23-s + (0.309 − 0.951i)25-s + (0.690 + 0.224i)27-s + (0.5 − 1.53i)31-s + (0.118 + 0.363i)36-s + (−0.309 − 0.224i)45-s + (0.690 − 0.951i)48-s + (0.809 − 0.587i)49-s + ⋯ |
L(s) = 1 | + (0.690 − 0.951i)3-s − 4-s + (0.809 − 0.587i)5-s + (−0.118 − 0.363i)9-s + (−0.690 + 0.951i)12-s − 1.17i·15-s + 16-s + (−0.809 + 0.587i)20-s + (−1.11 − 1.53i)23-s + (0.309 − 0.951i)25-s + (0.690 + 0.224i)27-s + (0.5 − 1.53i)31-s + (0.118 + 0.363i)36-s + (−0.309 − 0.224i)45-s + (0.690 − 0.951i)48-s + (0.809 − 0.587i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.402200258\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.402200258\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + T^{2} \) |
| 3 | \( 1 + (-0.690 + 0.951i)T + (-0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.690 - 0.951i)T + (-0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \) |
| 71 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (0.690 + 0.951i)T + (-0.309 + 0.951i)T^{2} \) |
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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
−8.789196311636378909671791240525, −8.031670780878078941298781397010, −7.45999097625379653816638896750, −6.30266247705765858837862261649, −5.78140935916239904339208938486, −4.71064086354339178716757819084, −4.13525326074252702725267783869, −2.77554476653105212513223304734, −1.99594019166974171317470473737, −0.872422180501151396981450912879,
1.58994487346585357463703675229, 2.92792102344759143204084063296, 3.55616666127907302260845996365, 4.33090295424358252528808010628, 5.20522960142232574925200879046, 5.86998147348822712854221409593, 6.85849981349924291122143026155, 7.83237309387956506971414849077, 8.622155809846954676361557309933, 9.207985279863361696607592991370