L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s − 1.41·5-s + (−0.707 − 0.707i)8-s + (−1.00 + 1.00i)10-s − 1.41i·11-s − i·13-s − 1.00·16-s + 1.41i·20-s + (−1.00 − 1.00i)22-s + 1.00·25-s + (−0.707 − 0.707i)26-s + (−0.707 + 0.707i)32-s + (1.00 + 1.00i)40-s + 1.41i·41-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s − 1.41·5-s + (−0.707 − 0.707i)8-s + (−1.00 + 1.00i)10-s − 1.41i·11-s − i·13-s − 1.00·16-s + 1.41i·20-s + (−1.00 − 1.00i)22-s + 1.00·25-s + (−0.707 − 0.707i)26-s + (−0.707 + 0.707i)32-s + (1.00 + 1.00i)40-s + 1.41i·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.004161556\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.004161556\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 13 | \( 1 + iT \) |
good | 5 | \( 1 + 1.41T + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + 1.41iT - T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - 1.41iT - T^{2} \) |
| 43 | \( 1 - 2T + T^{2} \) |
| 47 | \( 1 - 1.41T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + 1.41iT - T^{2} \) |
| 61 | \( 1 - 2iT - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - 1.41T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + 2iT - T^{2} \) |
| 83 | \( 1 - 1.41iT - T^{2} \) |
| 89 | \( 1 + 1.41iT - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−10.30057413330274956547939167049, −9.140141055400086623878744426553, −8.265147077845028517519185684863, −7.51010560449237694889688679136, −6.27718945984503453067121096975, −5.46232096197144585343636146710, −4.37230122847585154368583145156, −3.53762661819863578054090412617, −2.81303690122289617830950363093, −0.799158611783977020561209261077,
2.35723789587639709201561126128, 3.82431174700028423177915923583, 4.27447883147947906189345472963, 5.18071417885716455389129283630, 6.47752045783961125306666720327, 7.28657436058804258769550037318, 7.68770928382644583689887366905, 8.712802123389830418130725114453, 9.499626245371266361180892271581, 10.84399028364160447427385692993